This chapter details the PR experimental apparatus, which includes the PR device itself, as well as the propellant flow and RF electrical circuits that are used in the laboratory or packaged as a miniaturised subsystem. The standard operating conditions are with
˙
m = 100SCCM of Ar and Vpwr = 300V at 13.56MHz, targeting P ∼ 5W for use on microspacecraft.
The motivation for CFD-plasma modelling stems from the challenges in obtaining precise experimental measurements at high spatial and temporal resolution within the constricted geometry of PR. Simulations are performed to complement experimentation, and work onPR progresses in a leapfrogging manner between these two techniques.
The PR simulation mesh replicates the different components in the physical device at actual scale. A hemispherical downstream region allows vacuum expansion to be modelled without computational anomalies. The CFD and CFD-plasma modelling techniques are out- lined, with a description of both the fluid and plasma numerical methods and their respective parameters. The convergence to the final solution is steady state in CFD simulations, and periodic in CFD-plasma simulations. The physical properties database is an integral part of the simulation program, and has been manually populated with highly precise data with a focus on temperature dependent parameters.
Slip flow
This chapter presents cold gas CFD simulations ofPocket Rocket performed in the rarefied slip flow regime [84]. In this regime, boundary layer effects are dominant, and have a sig- nificant influence on the overall flow behaviour of the system. The correct flow boundary conditions must be used in the CFD simulations in order to produce results that are physic- ally accurate. Theoretical concepts of rarefied flow dynamics are introduced, followed by a detailed discussion of the slip regime boundary condition and the accommodation coefficients used to implement slip flow. Different flow boundary conditions are tested in a total of 54 CFD simulations performed with two variations of the Pocket Rocket geometry (PR-A and PR-Z), and using Ar and N2 propellants. The CFD simulation results are compared against experimental measurements to confirm the accuracy of the slip boundary condition across all the tested operating conditions.
3.1
Rarefied gas dynamics
3.1.1 Mean free path
The mean free pathλis the statistical mean distance travelled by a moving particle between
collisions with other moving particles. In a fluid medium,λ may be characterised using its
dynamic viscosityµ: λµ= µ p r πkBT 2m (3.1)
wherekBis the Boltzmann constant, pandT are the static pressure and temperature of the fluid, andm is the molecular mass of the constituent fluid molecule.
With sufficient rarefaction, gases begin to act more like individual particles and less like 37
a cohesive fluid, and the fluid description ofλµ ceases to be accurate. A kinematic relation
is required, characterised using the Lennard-Jones collision diameter σ of the gas molecule
[82]:
λσ = √kBT
2πpσ2 (3.2)
λσ preserves its accuracy even in rarefied conditions, and is therefore more appropriate for
the present study of PR in the . 1Torr range. Hereafter, the subscript ‘σ’ is dropped for
convenience, except when it is necessary to distinguish between λµ and λσ. As a general estimate,p∼1Torr andT ∼300K givesλ∼0.06mm.
3.1.2 Knudsen number
The Knudsen number Kn is a dimensionless parameter which determines whether a flow is better characterised by continuum or statistical mechanics. It is defined as the ratio of the mean free pathλto the characteristic length Lof the flow system:
Kn= λ
L (3.3)
In general, L is the smallest length scale in a particular geometry. For a cylinder, L is
typically its radius, unless its length is smaller than its diameter, in which caseLis defined
to be half the length.
For the PRgeometry (Figure2.8), the respective values are specified to beLP = 6mm in the plenum using the latter definition, whileLC= 2.1mm in the discharge chamber using the former definition. Assumingp∼1Torr andT ∼300K, this gives Kn∼0.03in the discharge chamber. The value ofLD = 22.5mm is selected for the downstream region according to the dimensions of the glass expansion tube using the former definition. The smaller radius of the glass expansion tube sets a stricter criteria for assessing the validity of the solution in the downstream region for low pressure scenarios.
3.1.3 Flow regimes
Flow can be categorised into four regimes according to their respective Kn range: continuum (Kn . 0.01), slip (0.01 . Kn . 0.1), transitional (0.1 . Kn . 10), and free molecular (Kn&10).
Continuum flow is dominated by fluid viscosity. Hagen-Poiseuille flow is an example of continuum flow. It is the laminar flow of an incompressible viscous fluid through a high aspect ratio channel induced by a pressure difference between upstream and downstream. The axial
velocity of the fluid in the channel has a peaked parabolic profile, as the fluid in the middle of the channel is moving the fastest while the fluid in contact with the channel wall is stationary with respect to the wall. This observation of the stationary fluid boundary is embodied in the no slip boundary condition. Another continuum flow example that prominently features the no slip boundary condition is Couette flow, where an incompressible viscous fluid flows between two surfaces, with one surface moving tangentially relative to the other. The no slip boundary condition at each surface forces the fluid to shear, resulting in a linear velocity profile between the two surfaces.
In the rarefied slip regime,λbecomes comparable toL, and boundary layer effects begin
to dominate flow behaviour. The no slip boundary condition is no longer valid for flow systems where Kn & 0.01, which is the case in the discharge chamber of PR. For accurate results, the boundary layer must be treated accordingly using the slip boundary condition.
Free molecular flow occurs in near-vacuum conditions where λ becomes very large. At
this Kn scale, gas molecules rarely encounter one another. They resemble discrete particles, travelling in straight lines and interacting primarily with the walls of the flow system. Con- sequently, fluid concepts like pressure and viscosity no longer apply. For example, the inter- stellar medium exists in the state of free molecular flow. In terrestrial laboratories, the free molecular regime may be attained in vacuum chambers through turbomolecular or cryogenic pumping.
Generally, fluid mechanics break down upon entering the transitional regime. Hence, CFD simulations are limited to modelling continuum flow, and the treatment of rarefied flows with higher Knudsen numbers require modelling techniques such as molecular dynam- ics (MD) or direct simulation Monte Carlo (DSMC). However, MD simulations are extremely computationally expensive for all but very small systems since it scales by the square of the number of molecules involved. A more practical solution is DSMC simulations which use a characteristic particle to represent a large ensemble of real molecules. While this reduces the scale of the problem, DSMC simulations are nevertheless still very much more compu- tationally expensive than CFD simulations for modelling weakly rarefied flows, particularly in the continuum and slip regimes. Studies seeking to model flows spanning a wide range of Knudsen numbers typically resort to a hybrid CFD/DSMC approach, applying CFD mod- elling techniques to regions of low Kn, and using the results thus obtained as a boundary condition for the neighbouring regions of high Kn where DSMC modelling techniques are employed [85].