0 100 200 300 400 500 600
0 0.2 0.4 0.6 0.8 1
R* [-]
t/tend [-]
RPE + T-eqn.
λvap λvap-10%
λvap+10%
(a) HKR, C5-90.
0 100 200 300 400 500 600
0 0.2 0.4 0.6 0.8 1
R* [-]
t/tend [-]
RPE + T-eqn.
λvap λvap-10%
λvap+10%
(b) HKR, C5-120.
0 100 200 300 400 500 600
0 0.2 0.4 0.6 0.8 1
R* [-]
t/tend [-]
RPE + T-eqn.
Cδ
Cδ-10%
Cδ+10%
(c) SHFM, C5-90.
0 100 200 300 400 500 600
0 0.2 0.4 0.6 0.8 1
R* [-]
t/tend [-]
RPE + T-eqn.
Cδ
Cδ-10%
Cδ+10%
(d) SHFM, C5-120.
Figure 6.12: Sensitivity of the bubble radius to a 10% change in the model coefficient.
All cases use R˚0 “50.
The errorbars in Fig. 6.13 indicate the deviation for the calibrated coefficients of the HKR for different mesh and time resolutions. The diagram shows the errorbars for R0˚ “5 and R˚0 “25 as well as Rp “5 and Rp “10 as a function of time. The influence of the numerical parameters is small compared to the change ofλvap with T0, Rp and R0˚. Therefore, the calibration of the model coefficients represents an adaption with respect to the physical conditions and it is not significantly affected by the numerics.
6.3. SENSITIVITY ANALYSIS FOR THE MODEL COEFFICIENTS AND DNS
RESULTS 104
0.04 0.05 0.06 0.07 0.08 0.09 0.1
85 90 95 100 105 110 115 120 125 λvap [-]
T0 [K]
Rp=5, R*0=5 Rp=5, R*0=25 Rp=10, R*0=5 Rp=10, R*0=25
Figure 6.13: Influence of the numerical setup (mesh and time resolution, initial pressure) to the calibrated coefficients of the HKR.
Bubble expansion in superheated jets
Chapters 5 and 6 discuss single bubble growth as the essential reference case and its simulation with the DG solver. Flash boiling is, however, a phenomenon where multiple bubbles occur. Findings based on single bubble models and simulations are not necessarily applicable to cases with multiple bubbles. Therefore, the present chapter studies the growth of interacting bubbles and compares the resulting growth rates to the cases of Ch. 6. The setups of Ch. 7represent a section of a superheated liquid jet and are introduced in Sec. 7.1. Section 7.2 provides the results of the simulations with multiple bubbles and the comparison to the single bubble cases.
Finally, Sec. 7.3 suggests closure conditions for the LES of flash boiling jets which are based on the present computations.
7.1 General setup
Section2.1.2discusses the role of bubble nucleation for flash boiling and summarises some of its fundamental aspects. The nucleation rate determines the bubble num-ber density after a certain time interval. The present work does not include the nucleation process itself but uses the bubble number density as a free parameter characterising the bubble distance. The bubbles are arranged in regular arrays such that equal bubble growth of all bubbles would result in a close packing configura-tion at the end of the computaconfigura-tions. The uniform bubble distance is Dbub and the
105
7.1. GENERAL SETUP 106
resulting distances in the Cartesian reference frame are1 Dx,bub “Dbub,
Dy,bub “ Dbub
?2 , Dz,bub “ Dbub?
6
3 . (7.1)
The distance between the bubbles is defined as a multiple of the initial radius with
Dbub “ΨbubR0, (7.2)
using Ψbub “ 10. The bubble number density is directly related to the bubble distance by Nbub,dens “1{pDx,bubDy,bubDz,bubq. The number densities can be varied by either modifying Ψbub or R0. Using i.e. Ψbub ă 10 leads to jet break-up in the very early stages of bubble growth, the dynamics of the growth process itself seem less significant and modelling of the spray at the nozzle exit as droplet clouds without primary break-up may be favourable. Instead, the bubble number density is modified by varying R˚0 “R0{Rcrit by a factor of 50 which leads to variations in bubble number densities by more than five orders of magnitude. Note that for large multiples of initial bubble radius, i.e. Ψbub "10, the early stages of bubble growth will not increase the volume of the jet. This can be shown analytically by [26]
V˚ “ Vjet,end
Vjet,0 “1`
?3 π 0.75
1
Ψ3bubpR˚3´1qp1´ρ˚q (7.3) which describes the ratio of a cylindrical volume after and before bubble growth.
The derivation of this equation is given in Appendix F. Figure 7.1 depicts V˚ as a function of the dimensionless bubble radius R˚ for different initial bubble spacings Ψbub. Starting from the critical radius at R0˚ “ 1 the jet volume will not increase as long as the condition ΨRbub˚ ą 10 holds. This, again, supports the omission of early stages of bubble growth and justifies the variation of the bubble number den-sity by changingR˚0 only. The parameter Ψbubis kept constant throughout this work.
1A close-packing of spheres may be defined by an array consisting of regular tetrahedrons and the center of the spheres would be located at the vertices of the tetrahedrons. If one edge of the tetrahedron is parallel to the x-direction in a Cartesian reference frame, the distance of the vertices in y-direction can be defined as the hight of a regular triangle, i.e. Dy,bub“?
3{2Dbub. Here, the distance in y-direction is Dy,bub“1{?
2Dbubă?
3{2Dbubresulting in an increased bubble count in y-direction by 20%.
1 2 3 4 5 6 7 8 9 10
1 10 100 1000
V* [-]
R* [-]
Ψbub=10 Ψbub=100 Ψbub=1000
Figure 7.1: Ratio between final jet volume and initial jet volume according to Eq.
(7.3). If the bubble distance is large compared to the initial radius, i.e. Ψbub is large, then the early stages of bubble growth do not contribute to the expansion of the liquid jet.
A typical nozzle diameter for the experiments at DLR [58] is Dnoz “ 0.5 mm.
Resolving the bubbles of the critical size atT0 “100 K andRp “5 (Rcrit “0.1µm) with 8 FV sub-cells across its diameter would require 20000 cells along the nozzle radius alone rendering a fully three-dimensional computation unfeasible. Therefore, the geometric configuration should be simplified and errors associated with the simplification should be assessed. As a first approximation the jet is represented by a column with one row of fully resolved bubbles in its center. Figure 7.2 shows a sketch of the computational domain. Note the orientation of the coordinate system with its origin positioned at the gas-liquid interface. Symmetry boundary conditions in the y- and z-directions as well as in the positive x-direction mimic the close packing of bubbles. In the negative x-direction a low pressure vapour reservoir allows for the expansion of the liquid column. The different fluid states are the same as described in Fig. 2.3: The liquid is superheated at state B and it contains saturated vapour bubbles at state C. The pressure in the vapour reservoir (state D) is below the saturation pressure for the local temperature. This simplified setup includes essential physical aspects of flashing flows and allows to conduct first qualitative and quantitative investigations at reduced computational cost. This setup is referred to as Setup 1 in the remainder of this work. For Setup 2 the approximation of the jet’s geometry is improved. The liquid bulk is now represented by a cylindrical slice which can expand to the negative x- and y-directions, while the z-directions and the positive x- and y-directions are treated as symmetry planes, see the illustration on the top right in Fig. 7.2. This setup is expected to represent fluid within the jet and away from the leading edge. Setup 3 is depicted in the
7.1. GENERAL SETUP 108 bottom right of Fig. 7.2, it additionally allows for an expansion of the liquid to the negative z-direction and represents fluid at the leading tip. The representation of the actual jet geometry improves from Setup 1 to Setup 3 but the computational effort also increases. The deviations in the bubble growth rates for the different geometries and the suitability of Setup 1 to represent jet expansion are discussed in Sec. 7.2.2.
x y z
Figure 7.2: Different geometrical setups for the investigation of multiple bubble growth in superheated liquids. The fluid states B, C and D are labelled according to Fig. 2.3.
A second important aspect of the configuration is the thickness of the liquid layer, i.e. Lx,liq in Setup 1,Lx,liq andLy,liq in Setup 2 and additionallyLz,liq in Setup 3. The thickness Lx,liq is defined by the number of bubbles in x-direction and the bubble distance by
Lx,liq “Nx,bub¨Dx,bub. (7.4)
In Setup 1, the thicknesses in y- and z-directions are given by Ly,liq “ 2Dy,bub and Lz,liq “ 2Dz,bub, respectively. The dimensions of Setup 2 and 3 are set to Ly,liq “ Lx,liq and Lz,liq “ Ly,liq “ Lx,liq. The bubble column in Fig. 7.2 contains, e.g., five bubbles in x-direction. The size of the low pressure reservoir is defined by Lx,buf “ 0.5¨Lx,bub. Setup 1 is the reference for analysing the dependency of the
bubble growth rates on the thickness of the layer, i.e. on the number of bubbles resolved in a certain spatial direction. These results are also discussed in Sec. 7.2.2.
Table 7.1 summarises the default geometrical and numerical parameters. The mesh is equidistant and resolves the initial bubble radius with four FV sub-cells.
Thus, the total number of cells will not depend on the physical parameters T0, Rp and R˚0 but only on the geometrical configuration and the number of bubble layers.
A sensitivity study varying the parameters in Tab. 7.1 shows a small influence on the numerical parameters, only. An increase of the mesh resolution or of the buffer zone size changed the resulting bubble radii by less than five percent. The CFL condition has a larger effect only for very small CFL numbers. However, the smaller CFL numbers caused larger mass losses as the level-set approach and the ghost fluid method used in the interface Riemann solver are inherently not mass conserving.
With CF L “ 0.1 mass losses have been of the order of five percent while it has been around one percent with CF L “ 0.9 and CF L “ 0.9 is therefore used. The investigation of multiple bubble growth uses the HKR as given in Eq. (3.20) or the SHFM as in Eqs. (3.22) and (3.23) to model vaporisation. The model coefficients are taken from the investigation in Ch. 6, see also Appendix D.
Table 7.1: Geometrical and numerical parameters.
Parameter CFL R0{∆x Ψbuffer Ψbub
Value 0.9 4 0.5 10