Quiescent Flux, or just “Queef,” follows from the definition of continuous jets:
SJs, similar to continuous jets (CJs), are studied by examining changes in the fluid velocity, density, temperature, and concentration of the component fluids both in the jet and in the ambient fluid. The readers are encouraged to consult a few available review articles for general experimental and simulation observations with SJs including the article by Glezer and Amitay (2002). In this chapter, we explore analytical models of SJs for two- dimensional (2D) plane SJs and axisymmetric SJs in cylindrical and spherical coordinates. The models start with the most simplified case for an SJ in a quiescent and infinitely large environment. We then gradually introduce more complex background flows (including coflow and crossflow) or interaction with a wall in the case of a wall SJ. The main results are based on a series of publications (Krishnan and Mohseni 2009a, 2009b, 2010; Xia and Mohseni 2010, 2011). We start by a brief discussion of the SJ actuators, the experimental setup, and a model of the actuator
A nondimensional stroke ratio (L/d) and the Reynolds number (Re) have beenidentified as key actuator operational parameters that influence an SJ (Smith and Glezer 1998). For the cavity-diaphragm setup used in this experiment,they are obtained from an incompressible flow model, where it is assumed that the volume displaced by the membrane is equal to the volume ejected from the orifice (Figure 1.4). As a result, it is quite important to measure the piezodiaphragm displacement during the operation ofthe actuatorto properly estimate the stroke ratio of the ejecting jet from the actuator. This has been achieved with the laser setup described earlier (see Krishnan and Mohseni, 2009a, for details).
Reynolds number. The jet velocity, defined by the Reynolds number,
R = uL / ν = puL / μ
where:
ρ is the density of the fluid (SI units: kg/m3)
u is the flow speed (m/s)
L is a characteristic length (m)
μ is the dynamic viscosity of the fluid (Pa·s or N·s/m2 or kg/(m·s))
ν is the kinematic viscosity of the fluid (m2/s).
is crucial in jet impingement processes since it strongly influences the local, line-averaged, and area-averaged Nusselt number
The Reynolds number is therefore explicitly seen to vary with both membrane driving frequency and amplitude, with the stroke ratio appearing to be independent of frequency. This independence of stroke ratio on frequency is not accurate as the use of a piezoelectric diaphragm as a driver gives rise to a coupling between the frequency and deflection, and consequently the stroke ratio. However, for purposes of calculating the jet parameters, the simple model described earlier serves the purpose. In summary, Equations 1.4 and 1.5 express the dependency of the critical actuator parameters on the diaphragm driving frequency and deflection amplitude.
Incompressibility Check* We are now in the position to verify the validity of the incompressibility assumptionused in the preceding calculations. In the following experiments, the actuation frequency is of the order of 1000 Hz and L is of an approximate magnitude of 0.01 m. Therefore, the velocity is of the order of 10 m/s.As a result,the operational Mach number for the actuators is quite below 0.3 that makes the flow virtually incompressible.
Eddy Viscosity Approach to Modeling SJs The last century has seen a significant effort in the scientific community in developing models for the averaged flow properties of laminar and turbulent CJs. An elegant and strikingly simple solution for modeling a laminar continuos jet was offered by Schlichting (1933). He found that an axisymmetric laminar jet could have a self-similar solution. He predicted a linear spreading rate for the jet, where the spreading rate was controlled by the fluid viscosity. This result was later extended to axisymmetric turbulent jets by supplementing the kinematic viscosity with an eddy viscosity to compensate for the enhanced momentum mixing in a turbulent CJ (Schlichting 1979). The enhanced eddy viscosity could be easily calculated from the jet spreading rate.
In a similar fashion, it has been recently hypothesized (Krishnan and Mohseni 2009a, 2009b) that the enhanced mixing and spreading rate of an SJ above what has been observed for a related turbulent CJ could be attributed to an enhanced eddy viscosity due to the pulsatile nature of an SJ.
To obtain the volume displaced by the membrane, the shape of the deflected membrane and the amplitude at the center of the membrane are required (Krishnan and Mohseni 2009a). The shape is obtained from the classical theory of plates (Timoshenko 1999)