1 Numerical simulations of volcanic jets: Importance of vent overpressure 1 Darcy E. Ogden 1* , Kenneth H. Wohletz 2 , Gary A. Glatzmaier 1 & Emily E. Brodsky 1 2 1 Earth & Planetary Sciences Department, University of California at Santa Cruz, 1156 High Street, Santa Cruz, CA 3 95064 4 2 Los Alamos National Laboratory, EES11 MS F665, Los Alamos, NM 87545 5 * corresponding author, [email protected]6 7
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Numerical simulations of volcanic jets: Importance of vent overpressure 1
Darcy E. Ogden1*, Kenneth H. Wohletz2, Gary A. Glatzmaier1 & Emily E. Brodsky1 2
1 Earth & Planetary Sciences Department, University of California at Santa Cruz, 1156 High Street, Santa Cruz, CA 3 95064 4 2 Los Alamos National Laboratory, EES11 MS F665, Los Alamos, NM 87545 5 *corresponding author, [email protected] 6
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2
Abstract 7
Explosive volcanic eruption columns are generally subdivided into a lower, momentum-8
dominated or gas-thrust section and an upper convection-dominated plume. Where pressure in 9
excess of atmospheric exists within the vent, the gas-thrust region is overpressured and develops 10
a jet-like structure of standing and reflecting shock waves known as a Mach Stem. Using a 11
pseudogas approximation for a mixture of rhyolite tephra and gas, we conducted a series of 12
numerical experiments that simulate the effects of Mach Stem development on the gas-thrust 13
region. Our results show that the strength and position of standing shock waves are strongly 14
dependent on the eruptive fluid pressure and vent radius. These factors control the gas-thrust 15
region’s shape and height and the character of vertical heat and mass flux into the convective 16
plume. With increased overpressure, the gas-thrust region becomes wider and develops an outer 17
sheath or shear zone in which the erupted mixture moves at higher speeds than it does near the 18
center of the column. The radius of this sheath is linearly dependent on the vent radius and the 19
square root of the overpressure. The sheath structure results in an annular vertical velocity (and 20
vertical heat flux) profile at the base of the convective plume. This result is in stark contrast to 21
the generally applied Gaussian or top-hat profile, which has been a fundamental assumption for 22
calculation of buoyant plume rise within and dispersion from an eruption column. 23
24
25 26
3
Notation 26
c Sound speed (m s-1). 27
CP Specific heat capacity of a gas at constant pressure (J kg-1K-1). 28 CV Specific heat capacity of a gas at constant volume (J kg-1K-1). 29
CS Specific heat capacity of a solid (J kg-1K-1). 30 hdisk Mach disk height (m). Height on jet axis of the standing Mach disk relative to vent exit. 31
K Overpressure ratio. Ratio of vent pressure (Pvent) to atmospheric pressure (Patm). 32 M Mach number. Ratio of fluid velocity (v) to sound speed in the fluid (c). Subscript: vent- 33
M at the vent, 1- just before the Mach disk, 2- just after the Mach disk. 34 n Mass fraction solids in the pseudogas mixture. 35
P Pressure (Pa). Subscript: vent- pressure at vent exit, atm- atmospheric pressure and 36 initial pressure in simulation box 37
q Vertical convective heat flux (J m-2 s-1). Subscript: vent- heat flux at the vent, peak- 38 highest heat flux in a horizontal slice located at z=1.25hdisk 39
Q Vertical convective heat flow (J s-1), i.e. heat flux integrated over an area. 40 r Radius (m), i.e. distance from jet centerline. Subscript: vent- radius of the vent, 41
max- radius of the simulation mesh, peak- horizontal location of
!
˙ q peak relative to jet 42 centerline 43
R Specific gas constant (J kg-1 K-1). Subscript: gas- of the gas portion of the pseudogas, 44 fluid- of the pseudogas itself. 45
Re Reynolds number. Re=ULν-1 where L is a characteristic length scale (e.g., the radius of 46 the jet), U a characteristic velocity, and ν the viscous diffusivity of the eruptive fluid. 47
T Temperature (K). Subscript: vent- at vent, atm- initial temperature in the mesh, fluid- 48 temperature of the eruptive pseudogas 49
v Velocity (m s-1). Subscript: r- radial component, z- vertical component, vent- at vent (all 50 vertical). 51
w Approximate thickness of fast moving sheath above the Mach disk (m). 52
z Vertical distance relative to vent exit (m), i.e. height. Subscript: max- height of 53 computational mesh. 54
γ Isentropic expansion coefficient for an ideal gas (γ=Cp/Cv). 55
Γ Isentropic expansion coefficient for a pseudogas. 56
ρ Density (kg m-3) Subscript: vent- at the vent. 57
σh Standard deviation of the Mach disk height (hdisk) over time. 58 59
4
1. Introduction 59
In large, explosive volcanic eruptions, the eruptive fluid issues from the vent as a high 60
speed, compressible gas with entrained solid particulates. It is important to quantify the behavior 61
of this gas-thrust region because it provides a connection between the fluid dynamics in the 62
conduit and that of the buoyant column. If the eruptive fluid velocity is at or greater than sonic 63
and vent pressure is higher than atmospheric pressure, the dynamics will be complicated by the 64
presence of standing shock waves that can drastically alter the distribution of the vertical heat 65
flux necessary for eruption column stability. The fluid dynamics and structure of a compressible 66
jet issuing from a sonic nozzle into an ambient atmosphere of lower pressure are well known 67
from experimental, analytical and computational studies (Figure 1,[Thompson, 1972]). Although 68
application of compressible jet dynamics to explosive volcanic eruptions was first suggested over 69
25 years ago by Kieffer [1981], the concept has yet to be widely applied in modeling and 70
analysis of explosive eruption columns. In this paper, we present computational results that 71
quantify the important effects of vent pressure on the fluid dynamics of volcanic jets and show 72
that overpressured jets produce vertical heat flux profiles that are drastically different than those 73
of pressure-balanced jets. (Note: to avoid confusion, here we use the physics convention and 74
consider “heat flux” the thermal energy transfer per area per unit time (J m-2 s-1) and “heat flow” 75
the thermal energy transfer integrated over an entire area per time (J s-1). In volcanology 76
literature, the term “heat flux” is often used to mean either of these things [e.g. Woods, 1988; 77
Mastin, 2007]). 78
Fixing the vent pressure to that of the surrounding atmosphere drastically limits the 79
complexity of the flow field. However, this simplifying assumption allowed fundamental 80
aspects of volcanic eruption dynamics to be determined. We briefly highlight only a few of them 81
5
here (for a more thorough review, see Valentine[1998] and Woods[1998]). Wilson et al. [1978] 82
demonstrate that eruptive columns behave like convective thermal plumes [Morton et al., 1956] 83
with a height proportional to the quarter-root of mass flow. This important finding built upon the 84
earlier prediction that eruptive mass flow is determined by conduit and vent shape and size 85
[Wilson, 1976]. Since that time, 30 years of observation and experiments have supported 86
Wilson’s predictions and demonstrated that vent shape and size also evolve during the course of 87
an eruption [Wohletz et al., 1984; Woods, 1995; Valentine and Groves, 1996; Sparks et al., 88
1997]. Notably in the last decade, a research group at the University of Pisa [e.g. Dobran, 1992; 89
Macedonio et al., 1994; Papale et al., 1998; Neri et al., 2007] demonstrated the advantages of 90
making nonlinear numerical simulations in two and three dimensions with sophisticated 91
constitutive relationships and subgrid-scale turbulence parameterizations. These models have 92
improved our understanding of the factors that influence eruptions including magma viscosity, 93
rheology and composition. 94
Kieffer first suggested that the well-understood fluid dynamics of an overpressured, 95
supersonic jet could describe volcanic eruption dynamics; she used this understanding to explain 96
the lateral blast of Mount St. Helens on May 18, 1980 [Kieffer, 1981]. She applied the method of 97
characteristics to solve for the general flow field and locations of standing shock structures in the 98
proximal areas of the blast (for a description of this commonly used method, see [Thompson, 99
1972]). This approach was later expanded [Kieffer, 1989] and used with laboratory experiments 100
of overpressured jets as volcanic analogs [Kieffer and Sturtevant, 1984]. This theory of volcanic 101
jets has been more widely applied in the planetary volcanism regime, e.g. sulfuric plumes on Io 102
[Kieffer, 1982], where large overpressure ratios are more easily achieved due to low atmospheric 103
densities. However, the theory that the gas-thrust region of volcanic jets may controlled by 104
6
compressibility effects (and associated shock waves) has not been widely applied to eruptions on 105
Earth. 106
Despite the lack of extensive application and quantification, there have been a few 107
computational and analytical studies, some with overpressure effects. Wohletz et al. [1984] 108
applied an implicit multifield Eulerian numerical method to simulate the initial phases of a large 109
caldera-eruption, showing that overpressure produces a blast wave analogous to the physics of a 110
temperature (c), vertical velocity (d), vertical mass flux (e), and density (f) for a typical 853
overpressured simulation. The vertical velocity profile clearly has a larger influence on the 854
vertical heat flux profile than the temperature, density, or fluid distribution. The turbulent nature 855
of the simulations can also be seen. Snapshots are of area of interest at t=20s for the jet with 856
rvent=40m and K=20. 857
858
Figure 9. 3D visualization of time-averaged top hat (a) and annular (b) vertical heat flux 859
profiles above the Mach disk rotated about the z-axis. x- and y-axes are horizontal spatial 860
axes extrapolated from radial profile. Shading and grid scale the same for both simulations. Fig. 861
(a), a top-hat profile vertical heat flux profile of simulation rvent=10m and K=1, is the expected 862
profile for K=1 or incompressible fluids. Fig. (b), an annular vertical heat flux profile of 863
simulation rvent=10m and K=5, is typical of all simulations with K>1. Note the hollow center in 864
(b) and flat top of (a). The heat flux relative to the vent heat flux in overpressured jets (b) is 865
lower than that of pressure balanced jets due to the increased area of the plume. 866
867
Figure 10. Time-averaged vertical heat flux above the Mach disk as a function of distance 868
from the jet centerline for all simulations. Vertical heat flux (q& , Jm-2s-1) above the Mach disk 869
as a function of distance from the jet centerline (r) normalized to the vent radius (rvent). Each 870
frame is a different overpressure value showing the shape of the heat profile is a function of 871
overpressure. The increasing radius of the peak heat flux and the decrease in heat flux with 872
increasing overpressure are shown in Figures 11 and 12, respectively. (Note that the density and 873
therefore the heat flux at the vent increases directly with overpressure. Therefore, the apparent 874
45
increase in heat flux between K=1 and K=5 is actually due to the increase in heat flux at the vent. 875
Figure 9 shows these same profiles normalized to vent heat flux.) 876
877
Figure 11. Time-averaged radius of peak vertical heat flux above the Mach disk for all 878
overpressured simulations. The distance of the peak vertical heat flux can be seen as a proxy 879
for plume width in these overpressured jets. The plume width increases linearly with vent radius 880
and the square root of the overpressure. The values plotted here are the radial values under the 881
peaks of the curves in Figure 10. Symbols are simulation data. Line corresponds to only to the 882
equation shown which represents the data trend but is not a calculated curve fit. 883
884
Figure 12. Time-averaged peak vertical heat flux above the Mach disk for all 885
overpressured simulations. (a) The peak vertical heat flux in the plume above the Mach disk 886
is severely decreased by the overpressure as the greater expansion spreads the jet over a greater 887
area. (b) qpeak is normalized to qvent in order to account for the increase in qvent with K. Symbols 888
are simulation data. Line corresponds to only to the equation shown which represents the data 889
trend but is not a calculated curve fit. 890
891
Figure 13. Schematic change in vertical heat flux profile. An illustration showing the 892
geometry of the vertical heat flux distribution at the vent (a) and above the Mach disk (b). As 893
described in the text, by empirical fit to our data and simple geometric analysis, the shape and 894
magnitude of the vertical heat flux distribution above the Mach disk as a function of 895
overpressure, vent radius, and vent vertical heat flux can be described by the three equations 896
given. 897
46
898
Figure 14. Standard deviation (σh) of Mach disk height on the axis as a function of average 899
Mach disk height for all overpressured simulations. Oscillations in Mach disk height are 900
common in overpressured jets and increase with increasing height of the Mach disk. This 901
phenomenon is related to the larger shear layers produced in larger jets and their interaction with 902
the Mach disk. This turbulence production can be quantified by the Reynolds number, which, 903
for these simulations is of the order 104. Symbols are simulation data. Line corresponds to only 904
to the equation shown which represents the data trend but is not a calculated curve fit. 905
906
Figure A1. Positive results of benchmark simulations of laboratory underexpanded jets. 907
Simulations based on the laboratory experiments of Lewis and Carlson [1964]. Line is a curve fit 908
of the empirical equation determined by Lewis and Carlson. Diamonds are time averaged data 909
from our simulations of air jets at the same scale and inflow velocity. Stars are simulation results 910
at higher resolution but smaller mesh height and width. Diamond simulations have a mesh that 911
is scaled to the Mach disk height in a similar way to the large scale jets. 912
913
47
913 Figure 1. Schematic underexpanded jet. As fluid flows from a nozzle at sonic velocities with 914 a pressure that is greater than the atmospheric pressure (a), the fluid undergoes Prandtl-Meyer 915 expansion, rapidly accelerating to high Mach numbers (b, c) and decreasing in pressure and 916 density. A continuous series of expansion waves form at the nozzle exit (d), which are reflected 917 as compression waves from the free surface at the jet flow boundary (e). These compression 918 waves coalesce to form a barrel shock (f) roughly parallel to the flow, and a standing shock wave 919 called a Mach disk (g) perpendicular to the flow. The high Mach number fluid crossing the 920 Mach disk undergoes an abrupt decrease in velocity to subsonic speeds (j), and increases in 921 pressure and density. The resulting fluid dynamics after the Mach disk consist of a slow moving 922 (subsonic) core surrounded by a fast moving (supersonic) shell with a turbulent eddy producing 923 shear layer, or slip line (i), dividing these regions. The length scales of this structure are 924 dependent on the nozzle diameter and the ratio of the inflow pressure to the ambient pressure and 925 are weakly dependent on the isentropic expansion coefficient of the fluid. [Thompson, 1972] 926
927
48
927 Figure 2. Schematic axial profiles of pressure and vertical velocity for an overpressured 928 supersonic jet. Jets with vent velocities of sonic or greater speeds and vent pressures greater 929 than atmospheric rapidly expand into the atmosphere, resulting in decompression and 930 acceleration (dashed lines). The axial pressure overshoots the atmospheric pressure. The fluid 931 continues accelerating until crossing the Mach disk where it decelerates to speeds below sonic 932 and increases pressure to roughly that of the atmosphere. For comparison, the dotted lines show 933 the approximation of Woods and Bower [1995] for free-jet decompression, which assumes that 934 expansion ends when the fluid reaches atmospheric pressure and no shock wave is present. Their 935 treatment results in axial velocities at the base of the plume that are orders of magnitude greater 936 than those predicted by overpressured jet theory. The vertical velocity within the plume in the 937 Woods & Bower model is assumed to be independent of the distance from the jet axis. In 938 contrast, the vertical velocity within an overpressured supersonic jet varies with distance from 939 the axis. Above the Mach disk, it is minimum on the axis and peaks near the outer boundary of 940 the plume. 941
942 943
49
943 Figure 3. Simulation design. Thirty-five time dependent simulations of overpressured volcanic 944 jets were run with CFDLib in a 2D cylindrical mesh with an axis of symmetry passing through 945 the jet centerline. We begin with a cylinder of air at rest and specify an outflow condition at the 946 top of the box, a small inflow boundary representing the vent at the base, and free-slip 947 boundaries everywhere else. An inflow of pseudogas at sonic speeds is specified at the base. 948 The simulations consist of seven different overpressure values at the vent (K = 1, 5, 10, 20, 40, 949 80, and 100), and five different vent radii (rvent = 10, 20, 40, 80, and 100m). The height (zmax) 950 and width (rmax) of the mesh are ten and five times the expected height of the first Mach disk, 951 respectively. We present data from the small subsection of the total mesh, the “analyzed region”, 952 in the remainder of this paper. 953
954
50
Overpressured jet (K=5) 954
955
956 957
Pressure balanced jet (K=1) 958
959
960 961
962 963 Figure 4. Vertical velocity snapshots for overpressured (a) and pressure balanced (b) jets. 964 These simulations highlight the difference in velocity profile between a jet erupting at 965
51
atmospheric pressure (b) and one undergoing rapid expansion from a modest overpressure of 5 966 times atmospheric (a). Both simulations have vent radii of 10 m. The snapshots are at the same 967 times for both simulations: 0.0625, 0.125, 0.21875, 0.3125, 0.4375, and 3.0 seconds. The full 968 velocity vectors are included at every 5 grid cells. The same color and vector length scale is 969 used for both simulations. The area shown is a subsection of the entire mesh just above the vent.970
52
971 Figure 5. Mach disk height as a function of overpressure for all overpressured simulations. 972 The time averaged Mach disk height on the jet axis normalized to vent radius is shown to be 973 directly dependent on the square root of the overpressure. Symbols are simulation data. Solid 974 curve is the analytical expression and not a best-fit curve. The overprediction at large values of 975 K is likely due to the Mach disk being pushed downward by turbulent eddies produced at high 976 overpressures. 977
978
53
978
979 Figure 6. Snapshots of axial profiles of pressure, vertical velocity, density, and vertical heat 980 flux for a typical overpresured simulation. These snapshots from a simulation with rvent=40 m 981 and K=20 clearly show the rapid expansion of the gas to pressure values well below atmospheric 982 before returning to approximately atmospheric values after the Mach disk. The stepwise change 983 in vertical velocity and pressure as the fluid crosses the shock wave can be seen in (a) and (d). 984 These snapshots show only a small portion of the total mesh just above the vent as the mesh 985 extends to 2.5 km. 986
987
54
987 Figure 7. Time-averaged decrease in vertical velocity across the Mach disk on the axis as a 988 function of overpressure for all overpressured simulations. As discussed in the text, 989 increasing overpressure increases axial velocities resulting in stronger Mach disk shocks and a 990 greater decrease in velocity across the shock. 991
992
55
(a) (b) 992 993
(c) (d) 994 995
(e) (f) 996
56
Figure 8. Snapshots of volume fraction eruptive fluid (a), vertical heat flux (b), 997 temperature (c), vertical velocity (d), vertical mass flux (e), and density (f) for a typical 998 overpressured simulation. The vertical velocity profile clearly has a larger influence on the 999 vertical heat flux profile than the temperature, density, or fluid distribution. The turbulent nature 1000 of the simulations can also be seen. Snapshots are of area of interest at t=20s for the jet with 1001 rvent=40m and K=20. 1002
Figure 9. 3D visualization of time-averaged top hat (a) and annular (b) vertical heat flux 1005 profiles above the Mach disk rotated about the z-axis. x- and y-axes are horizontal spatial 1006 axes extrapolated from radial profile. Shading and grid scale the same for both simulations. Fig. 1007 (a), a top-hat profile vertical heat flux profile of simulation rvent=10m and K=1, is the expected 1008 profile for K=1 or incompressible fluids. Fig. (b), an annular vertical heat flux profile of 1009 simulation rvent=10m and K=5, is typical of all simulations with K>1. Note the hollow center in 1010 (b) and flat top of (a). The heat flux relative to the vent heat flux in overpressured jets (b) is 1011 lower than that of pressure balanced jets due to the increased area of the plume. 1012
1013
58
1013
59
Figure 10. Time-averaged vertical heat flux above the Mach disk as a function of distance 1014 from the jet centerline for all simulations. Vertical heat flux (q& , Jm-2s-1) above the Mach disk 1015 as a function of distance from the jet centerline (r) normalized to the vent radius (rvent). Each 1016 frame is a different overpressure value showing the shape of the heat profile is a function of 1017 overpressure. The increasing radius of the peak heat flux and the decrease in heat flux with 1018 increasing overpressure are shown in Figures 11 and 12, respectively. (Note that the density and 1019 therefore the heat flux at the vent increases directly with overpressure. Therefore, the apparent 1020 increase in heat flux between K=1 and K=5 is actually due to the increase in heat flux at the vent. 1021 Figure 9 shows these same profiles normalized to vent heat flux.) 1022
1023
60
1023 Figure 11. Time-averaged radius of peak vertical heat flux above the Mach disk for all 1024 overpressured simulations. The distance of the peak vertical heat flux can be seen as a proxy 1025 for plume width in these overpressured jets. The plume width increases linearly with vent radius 1026 and the square root of the overpressure. The values plotted here are the radial values under the 1027 peaks of the curves in Figure 10. Symbols are simulation data. Line corresponds to only to the 1028 equation shown which represents the data trend but is not a calculated curve fit. 1029
1030
61
1030 Figure 12. Time-averaged peak vertical heat flux above the Mach disk for all 1031 overpressured simulations. (a) The peak vertical heat flux in the plume above the Mach disk 1032 is severely decreased by the overpressure as the greater expansion spreads the jet over a greater 1033 area. (b) qpeak is normalized to qvent in order to account for the increase in qvent with K. Symbols 1034 are simulation data. Line corresponds to only to the equation shown which represents the data 1035 trend but is not a calculated curve fit. 1036
1037
62
1037
Figure 13. Schematic change in vertical heat flux profile. An illustration showing the 1038 geometry of the vertical heat flux distribution at the vent (a) and above the Mach disk (b). As 1039 described in the text, by empirical fit to our data and simple geometric analysis, the shape and 1040 magnitude of the vertical heat flux distribution above the Mach disk as a function of 1041 overpressure, vent radius, and vent vertical heat flux can be described by the three equations 1042 given. 1043
1044
63
1044
1045
Figure 14. Standard deviation (σh) of Mach disk height on the axis as a function of average 1046 Mach disk height for all overpressured simulations. Oscillations in Mach disk height are 1047 common in overpressured jets and increase with increasing height of the Mach disk. This 1048 phenomenon is related to the larger shear layers produced in larger jets and their interaction with 1049 the Mach disk. This turbulence production can be quantified by the Reynolds number, which, 1050 for these simulations is of the order 104. Symbols are simulation data. Line corresponds to only 1051 to the equation shown which represents the data trend but is not a calculated curve fit. 1052
1053
64
1053
1054 Figure A1. Positive results of benchmark simulations of laboratory underexpanded jets. 1055 Simulations based on the laboratory experiments of Lewis and Carlson [1964]. Line is a curve fit 1056 of the empirical equation determined by Lewis and Carlson. Diamonds are time averaged data 1057 from our simulations of air jets at the same scale and inflow velocity. Stars are simulation results 1058 at higher resolution but smaller mesh height and width. Diamond simulations have a mesh that 1059 is scaled to the Mach disk height in a similar way to the large scale jets. 1060