Jet Impinging on a Spring-Loaded Plate

Problem Description

In this problem, a circular jet impacts on a spring-loaded flat plate. The flow changes direction when it reaches the plate, resulting in an analytical force of:(1)
F = ρ Q V MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOraiabg2da9iabeg8aYjaadgfacaWGwbaaaa@3B4B@
Where,

ρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdihaaa@37C9@ is fluid density. Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyuaaaa@36DE@ is volumetric flowrate. V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOvaaaa@36E3@ is the jet velocity.

Volumetric flow of a jet of diameter d is Q = πd2/4, where d is jet diameter. Assuming a spring with a stiffness of k, the above force will displace the plate by Δx = F/k.

To reduce the oscillations, the plate motion is critically damped using a linear damper. The damping coefficient is set to c = 2(kM)1/2, so the plate of mass M approaches its equilibrium position without overshooting.

Table 1 shows the parameters used for the simulation here. Based on these values, the analytical plate displacement in the jet direction is 0.01m.
Table 1. Simulation Parameters for Jet Impinging on a Plate
ρ [kg/m3] µ [Pa.s] V [m/s] k [N/m] c [N.s/m] d [m] M [kg]
1000 0.001 10 7068.58 531.74 0.03 10

Numerical Setup

The square plate of side length w = 0.09m is placed in the middle of a cubic computational domain with a side length of h = 0.12m. Discretized by 30 particles across diameter (dx = 0.001m), only the jet fluid phase is simulated. Figure 1 shows a schematic of the problem.


Figure 1. Jet, Plate and Spring-Damper Configuration Schematic

Results

In Figure 2, the plate reaches a constant displacement of 0.0989m between 0.4s and 0.5s, very close to the analytical value of 0.01m. While the force applied to the plate has some oscillations, the time averaged force on the plate in 0.1s to 0.5s interval is equal to 69.94N. The analytical value of the impact force is equal to 70.68N.

The graph on the left shown in Figure 2 shows the plate displacement versus time. The graph on the right shows the fluid force on the plate due to jet impact. The dashed line is the average force.


Figure 2. Results of Jet Impingement on the Spring-Loaded Plate