Numerical investigation of rarefied vortex loop formation due to shock wave diffraction with the use of rorticity
Summary (3 min read)
A. Background
- In the past two decades, many micro-satellites, e.g., CubeSats, have been sent into space for purposes such as earth observation, telecommunication, and navigation 33 .
- Non-electric propulsion systems (e.g. cold gas micro-thrusters) have been deployed extensively for orbit transfer and manoeuvring due to their high reliability 33 .
- In the continuum flow regime, shock wave diffraction and compressible vortex loop formation and development have attracted attention for decades.
- There are also some novel investigations of shock wave diffraction, such as diffraction from a curved exit from the shock tube 29 , the interaction of two perpendicular diffracting shock waves 44 and vortex merging caused by shock wave diffraction in three dimensions 6 .
B. Usage of rorticity
- This definition is ambiguous, and in some circumstances, the vorticity concentration can not be explained to be the existence of a vortex, e.g. in turbulent flow.
- It is unable to distinguish between rotational and irrotational structures, but both rotation and local shear movements can create vorticity.
- In Ref.47, the building of the vortex vector or rorticity is not derived from the continuum assumption of fluid flow so that rorticity is a purely mathematical concept and can be used in both continuum and rarefied flow.
- Since the new concepts are defined purely mathematically, and the calculation is based on linear algebra, there are no assumptions related to the fluid.
- Hence, the concepts are appropriate for all flow conditions.
1. Mesh and time-step independence
- The time derivative is discretised using a first-order implicit Euler scheme and a second-order central-upwind differencing scheme is applied to discretise the gradient terms, divergence terms, and diffusive terms, and also to interpolate the cell centre values to the cell faces.
- Since the fine mesh is not computationally expensive, and the spatial resolution is better, the fine mesh with a resolution of 0.25 mm is used in the remainder of this work.
III. SIMULATION DETAILS
- The working gas in all simulations of this work is nitrogen, and the non-dimensional parameters, including the Knudsen number and the Reynolds number, are based on the half-height of the shocktube, which is 0.01 m.
- The mean free path of the Knudsen number is calculated according to the conditions downstream of the primary shock wave inside the shock tube.
- The Rankine-Hugoniot relations are used to calculate the macroscopic parameters defined in each case of this work; the case setup is shown in Table II and III.
- In the dsmcFoamPlus solver, the NTC method is used for collision partner selection, and the variable hard sphere model with Larsen-Borgnakke energy redistribution (with an inverse rotational energy collision number of 5) is used to perform the collisions.
- The size of the domain outside the shock tube exit is varied for different cases because the computational costs are different.
IV. RESULTS AND DISCUSSION
- There is no visible shock structure within the vortex until the shock Mach number reaches 1.5.
- It it well-known that shock wave thickness increases with flow rarefaction and that the Navier-Stokes equations are unable to capture this physics.
- As illustrated in experiments 19 24 and simulations 53 31 , the vortex-induced shock pair occurs due to the high-velocity magnitude within the shear layer of the primary vortex loop in the continuum flow regime.
- As the Knudsen number increases, the embedded shock and the shear layer degenerate further.
- It can be concluded that the high rarefaction level will simplify the inner structures of the compressible vortex loop.
1. Velocity and pressure field comparisons
- An increase in local Knudsen number can be found in the primary shock, embedded shock, and the expansion at the exit.
- Through the comparison between Figures 8 and 9 , the difference of the results in Figure 8 between the two solvers can be explained.
- The axial pressure distribution profile differs between the slip flow and the higher Knudsen number regimes.
2. Rorticity and shear vector field
- The authors call the region formed by closed-loop streamlines the vortex atmosphere.
- Figures 11 and 12 present contours of the rorticity field and streamlines before and after the formation of the isolated rorticity region, or rorticity loop cross-section, with different shock Mach numbers, in the near continuum flow regime, as calculated using hy2Foam and dsmcFoamPlus.
- The high rorticity magnitude at the corner in the expansion fan indicates that a fluid element experiences a significant increase of rotational kinetic energy because the rorticity is defined as twice the fluid-rotational angular velocity, according to Equation (2.23) in Ref. 47.
- Such a phenomenon can also be confirmed from the algorithm validation in the DNS simulation result of a 2D Blasius-profile mixing layer flow in Ref. 47.
- The vortex centre, defined from the streamlines, is constrained within the circle of influence of the shear vector field, suggesting that the core of a vortex loop built on streamlines has both rotational and shear movements but the strength of the rotational movement is much stronger than that of the shear movement.
C. Evaluation of the rotational strength of the vortices
- With the introduction of rorticity, substituting Equation (1) into Equation ( 6), the following is obtained: EQUATION and thereby EQUATION ) Still, there are no strict rules on the limitation of this circuit size, resulting in the introduction of errors.
- The rorticity region within a vortex atmosphere is connected to the rorticity sheet at the tube exit (.
- The left-hand side of Equation ( 13) is the difference between the total shear vector flux and the remaining shear vector flux within a rorticity loop.
D. Geometrical characteristics of rarefied vortex loop
- Calculating the geometrical parameters of rarefied vortex loops is helpful in the understanding of the vortex loop coverage in the flow field.
- The authors can define the region with rorticity in the vortex described by streamlines as the rotational core of a vortex loop.
- This equivalent diameter intuitively describes the size of a circular ring's cross-section with the same area.
- When the shock Mach number is 1.4 and 1.3, a highly linear relationship between the equivalent radius of the rorticity ring and non-dimensional time is found.
- The expansion caused by low pressure outside the tube leads to growth of the vortex loop in the radial direction.
E. Failure of vortex loop formation
- Figure 23 displays the variation of the velocity and the streamlines due to the increase of rarefaction level with a shock Mach number of 1.6 in the slip flow regime.
- The vortex loop will be diluted during its propagation and should eventually disappear when the Knudsen number limit for formation is reached.
- Outside the tube, Figure 26 (b) presents the dimensionless tangential velocity profile calculated from DSMC at x =12 mm, which is 2 mm away from the tube exit, and t =0.12 ms.
- The thickness of the vortex sheet increases, and the tangential velocity difference decreases significantly with an increase of the Knudsen number.
- Typically, the Prandtl-Meyer expansion fan will not intrude into the shock tube, or the expansion will stay outside the tube exit.
V. CONCLUSIONS
- Attention has been primarily paid to the vortex loop formation caused by shock wave diffraction over a rectangular corner in dilute gas flows.
- Transient DSMC and compressible CFD simulations has been performed, and comparisons have been made between the results from the dsmcFoamPlus and hy2Foam where the Knudsen number allowed.
- An increase in flow rarefaction results in the innerstructure of vortex loops becoming simpler and the flow patterns in all the simulations of this work are laminar.
- The increase of rorticity flux in the isolated rorticity loop with time is attributed to the transformation from the shear vector to rorticity.
- Further investigation can be conducted to explore the relationship between the vortex formation and the Knudsen number.
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Frequently Asked Questions (12)
Q2. What is the effect of a thruster on the lip of the nozzle?
During the transient period as a thruster begins to fire, the sudden ejection of relatively high-pressure supersonic gas from a nozzle into a low-pressure environment generates a shock diffraction around the lip of the nozzle, resulting in lateral vortex formation.
Q3. What is the rorticity field of a vortex loop?
As the rorticity field of a vortex loop core is an irregular shape, an equivalent diameter of a vortex is defined asdeq = √ AR π18H , where t∗0 is the time that the rorticity loop formed and it is 0.25ms, 0.18ms, 0.15ms,0.13ms, 0.1ms for Ms = 1.3, Ms = 1.4, Ms = 1.5, Ms = 1.6, Ms = 2.0, respectively.
Q4. How many particles are required to reduce the statistical error in the computed collision rates?
The number of particles in each cell must be sufficient to reduce the statistical error in the computed collision rates; typically at least 20 particles per cell are required when using the no time counter (NTC) method to calculate the number of possible collision pairs42.
Q5. What is the effect of the shock Mach number on the vorticity?
Increasing the shock Mach number causes the shape of the rorticity loop cross-section to change comma-like shape to a mushroom shape.
Q6. What is the method for investigating rarefied flows?
The direct simulation Monte Carlo (DSMC) method3 is a standard tool for investigating rarefied flows with moderate to high Knudsen number.
Q7. What is the flow flux of a vortex?
The circulation can be decomposed into a rorticity flux that describes the fluid-rotational strength of a vortex and a shear vector flux that represents the shear strength of a vortex.
Q8. What is the main reason for the demand for thrusters in low Earth orbit?
The continuous growth of applications for cost-effective micro-satellites in low Earth orbit (LEO) is leading to a requirement for specialized thruster systems that can provide thrusts in the micro- and mili-Newton range, in order to control their motions and orbits8.
Q9. How many ensembles are required to obtain a 10% uncertainty in the velocity of nitrogen gas?
For instance, to obtain a 10% uncertainty in the velocity for nitrogen gas at a Mach number of 0.1, 286 ensembles with 25 particles in each cell are required.
Q10. What did the authors do to avoid reflections of expansion waves and incident shock waves?
The authors enlarged the fluid domain outside the shock tube exit to allow for a longer time scale and avoid reflections of expansion waves and incident shock waves.
Q11. What is the rorticity field of a vortex?
The rorticity field always coincides with the closed-loop streamlines if the vortex is stationary in the plane perpendicular to the rotational axis, such as in a Taylor-Green vortex sheet or a Burgers vortex, and it does not coincide with the closed-loop streamlines if the vortex translates.
Q12. What is the basic algorithm that all DSMC solvers follow?
An overview of the basic algorithm that all DSMC solvers follow can be given as:1. Update the position of all particles in the fluid domain using the particle tracking algorithm, which also deals with the motion of particles across faces of the mesh, and applies boundary conditions.