A numerical shallowwater model for gravity currents for a wide range of density differences
 Hiroyuki A. Shimizu^{1}Email author,
 Takehiro Koyaguchi^{1} and
 Yujiro J. Suzuki^{1}
DOI: 10.1186/s4064501701202
© The Author(s) 2017
Received: 21 July 2016
Accepted: 3 March 2017
Published: 28 March 2017
Abstract
Gravity currents with various contrasting densities play a role in mass transport in a number of geophysical situations. The ratio of the density of the current, ρ _{c}, to the density of the ambient fluid, ρ _{a}, can vary between 10^{0} and 10^{3}. In this paper, we present a numerical method of simulating gravity currents for a wide range of ρ _{c}/ρ _{a} using a shallowwater model. In the model, the effects of varying ρ _{c}/ρ _{a} are taken into account via the front condition (i.e., factors describing the balance between the driving pressure and the ambient resistance pressure at the flow front). Previously, two types of numerical models have been proposed to solve the front condition. These are referred to here as the Boundary Condition (BC) model and the Artificial Bed (AB) model. The front condition is calculated as a boundary condition at each time step in the BC model, whereas it is calculated by setting a thin artificial bed ahead of the front in the AB model. We assessed the BC and AB models by comparing their numerical results with the analytical results for a simple case of homogeneous currents. The results from the BC model agree well with the analytical results when ρ _{c}/ρ _{a}≲10^{2}, but the model tends to overestimate the speed of the front position when \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\). In contrast, the AB model generates good approximations of the analytical results for \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\), given a sufficiently small artificial bed thickness, but fails to reproduce the analytical results when ρ _{c}/ρ _{a}≲10^{2}. Therefore, we propose a numerical method in which the BC model is used for currents with ρ _{c}/ρ _{a}≲10^{2} and the AB model is used for currents with \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\).
Keywords
Gravity currents Numerical model Shallowwater model Front conditionIntroduction
Gravity currents are flows driven by density differences between the current and the ambient fluid. In geophysical settings, there are many types of highReynoldsnumber (typically \(\gtrsim 10^{3}\)) gravity currents that show a wide range of density ratios (ρ _{c}/ρ _{a}, where ρ _{c} and ρ _{a} are the densities of the current and ambient fluid, respectively), such as debris flows (ρ _{c}/ρ _{a}∼10^{3}; e.g., Iverson 1997), turbidity currents (ρ _{c}/ρ _{a}∼10^{0}; e.g., Meiburg and Kneller 2010), and pyroclastic density currents (ρ _{c}/ρ _{a}=10^{0}–10^{1} in the overlying parts and ρ _{c}/ρ _{a}=10^{2}–10^{3} in the underlying parts; e.g., Branney and Kokelaar 2002; Breard et al. 2016; Nield and Woods 2004). For the two extreme cases of ρ _{c}/ρ _{a}∼10^{0} and 10^{3}, the fluid dynamical features of gravity currents (e.g., the shape of the interface and the propagation of the flow front) have been studied in detail using experimental investigations (e.g., Marino et al. 2005; Martin and Moyce 1952; Dressler 1954; Rottman and Simpson 1983), numerical investigations (e.g., Cantero et al. 2007; Ooi et al. 2009), and theoretical modeling (e.g., Benjamin 1968; Hogg and Pritchard 2004; Huppert and Simpson 1980; Stoker 1992; Ungarish and Zemach 2005). For intermediate density ratios (10^{0}<ρ _{c}/ρ _{a}<10^{3}), there have been some previous studies (e.g., Birman et al. 2005; Bonometti et al. 2011; Gröbelbauer et al. 1993; Hallworth and Huppert 1998; Härtel et al. 2000; Ungarish 2007), but the dynamics of gravity currents within this density range is less well understood than that of the extreme cases.
The purpose of this study is to develop a numerical model of gravity currents for a wide range of ρ _{c}/ρ _{a} based on a shallowwater model. The shallowwater model is an efficient mathematical model that captures the essential features of the vertically averaged motion of gravity currents with free surfaces (see Ungarish 2009 for an extensive review). For simple initial and boundary conditions, analytical solutions of the shallowwater model for propagating gravity currents are available for a wide range of ρ _{c}/ρ _{a} (Ungarish 2007), and these analytical solutions have been verified by experimental measurements and direct numerical simulations using the Navier–Stokes equation (Bonometti and Balachandar 2010; Ungarish 2007). However, geophysical conditions of interest generally have rather complex initial and boundary conditions, so such analytical solutions are not always available. A numerical model that is applicable for complex initial and boundary conditions is highly desirable for simulations of gravity currents for a wide range of ρ _{c}/ρ _{a}.
This study is particularly concerned with a numerical treatment of the flow front of gravity currents. In the following sections, we formulate the mathematical problem and show that the numerical treatment at the flow front is key to correctly solving the dynamics of gravity currents for a wide range of ρ _{c}/ρ _{a} within the framework of the shallowwater model. We also assess previous numerical methods that have been used to calculate the behavior of the flow front by comparing numerical and analytical results, and we propose a numerical method to simulate the dynamics of gravity currents for a wide range of ρ _{c}/ρ _{a} under various geophysical conditions. Finally, as a geophysical application of our results, we develop a numerical model of a pyroclastic density current with strong density stratification.
Methods
Formulation
where h(x,t) is the local height and u(x,t) is the local horizontal velocity.
where Fr, which is an imposed frontal Froude number, is assumed to be a constant of order 10^{0} (e.g., \(\sqrt {2}\); Benjamin 1968).
Note that the density ratio ρ _{c}/ρ _{a} is included only in the front condition (7). Hence, to capture the effects of ρ _{c}/ρ _{a}, it is important to calculate the front condition correctly (Ungarish 2007).
Numerical methods
where Δ x is the constant cell length and Δ t is the time interval. F _{ i+1/2}, which is the intercell flux between cells i and i+1, is obtained by using an exact Riemann solver or an approximate Riemann solver, such as the Roe scheme (e.g., LeVeque 2002; Toro 2001). The time interval Δ t is limited by the Courant–Friedrichs–Lewy condition (e.g., LeVeque 2002; Toro 2001).
Boundary Condition (BC) model
The values of h _{N} and u _{N} are approximated by the values of h and u at the front cell (i.e., h _{ FC } and u _{ FC }).
Solving these three equations analytically (e.g., using Ferrari’s method for the solution of the quartic equation) or numerically (e.g., using the Newton–Raphson iteration method), we obtain \(h_{FC}^{n+1}\), \(u_{FC}^{n+1}\), and \(\Delta x_{FC}^{n+1}\), and hence, h _{N},u _{N}, and x _{N} at each time step.
Artificial Bed (AB) model
In the AB model, the conservation equations (Eqs. (6) and (8)) are numerically solved using a shockcapturing method for not only the interior, but also the outside of the current by a priori setting a thin artificial bed ahead of the front. Through this numerical procedure, the flow front is generated as the flow following a shock formed ahead of the front without any additional calculation (see Fig. 3 b). In this model, the thickness of the artificial bed (ε in Fig. 3 b) is the parameter that controls the front condition (i.e., the values of h _{N} and u _{N} for different values of ρ _{c}/ρ _{a}; see section 10.8 in Toro 2001).
Here, we analytically determined the relationship between ε and ρ _{c}/ρ _{a}, as well as that between u _{N} and ε, on the basis of the analytical solution for the slumping stage of the dambreak problem (e.g., LeVeque 2002; Toro 2001; Ungarish 2009). The initial conditions are h=1 and u=0 in the domain 0≤x≤1, and h=ε and u=0 in the domain x>1, at t=0. Let us consider the time evolution of the current before the rarefaction wave reaches the back wall x=0 (i.e., 0<t≤1).
from Eq. (14). The equation provides the relationship between h=h _{N} and u=u _{N} inside the current.
Note that because we use Eq. (15) here, these relationships (Fig. 5) are in the slumping stage.
after solving the equations of fluid motion (see section 13.12 in LeVeque 2002 for details).
Results and discussion
In this section, we compare the numerical results obtained from the BC and AB models with the analytical results, and assess the applicability of these models. Subsequently, as a geophysical application of our results, we develop a numerical model of pyroclastic density currents.
Comparison of analytical and numerical results
The good agreement in the results of the AB model for ρ _{c}/ρ _{a}=1000 reflects the fact that the dynamics of the gravity current becomes insensitive to the front condition for large values of ρ _{c}/ρ _{a}. In Fig. 5 b, ε approaches 0 as ρ _{c}/ρ _{a} increases. In the limit as ρ _{c}/ρ _{a}→∞ and ε→0,u _{N} asymptotically approaches its maximum value, 2, and h _{N} asymptotically approaches 0. For sufficiently small ε, the solution converges to that in the limit as u _{N}→2 and h _{N}→0, and it becomes insensitive to the value of ε (see Fig. 5 a). Indeed, as shown in Fig. 8 b, we can confirm that the result of the AB model with a very small ε(ε=1.0×10^{−10}) is indistinguishable from that for ρ _{c}/ρ _{a}=1000(ε=4.58×10^{−7}). According to Fig. 5, the results of the AB model for the dambreak problem are insensitive to ε when ε≲10^{−5}, which corresponds to \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\) (Fig. 5 b). Consequently, we suggest that the AB model is favorable for simulating gravity currents with high ρ _{c}/ρ _{a} for which the dynamics of the current is insensitive to the assumed value of ε.
Applicability of the BC and AB models
Our results indicate that the BC and AB models each have their own advantages and disadvantages. The results obtained from the BC model agree well with the analytical results when ρ _{c}/ρ _{a}≲10^{2} (Fig. 6 a), whereas they show a numerical oscillation at the flow front and tend to overestimate the front speed when \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\) (Fig. 7). No such numerical oscillation nor overestimation is observed in the results from the AB model. For currents with \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\), the AB model provides good approximations of the analytical results, given a sufficiently small ε (Figs. 5 and 8 b). For currents with ρ _{c}/ρ _{a}≲10^{2}, however, the AB model may fail to reproduce the analytical results for currents where the height and speed of the front change with time (Fig. 8 a). Accordingly, we propose that the BC model should be used for currents with ρ _{c}/ρ _{a}≲10^{2} and the AB model is applicable only to currents with \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\).
Geophysical application to pyroclastic density currents
Pyroclastic density currents (PDCs) are characterized by strong density stratification due to particle settling (e.g., Branney and Kokelaar 2002), whereby a dilute gravity current (particle suspension flow) with ρ _{c}/ρ _{a}=10^{0}–10^{1} overrides the dense basal gravity current (fluidized granular flow) with ρ _{c}/ρ _{a}=10^{2}–10^{3}. The dynamics of PDCs is complex because the dilute and dense currents are influenced by a number of physical processes such as particle settling (e.g., Bonnecaze et al. 1993), entrainment of ambient air (e.g., Johnson and Hogg 2013; Sher and Woods 2015), and basal resistance (e.g., Roche et al. 2008). In addition to the effects of these processes, our results suggest that the application of the correct numerical model to the flow front is important if we are to understand the dynamics and sedimentation of PDCs. The BC model should be applied to the overlying dilute current, while the AB model is applicable to the underlying dense current. Here, we discuss how the resistance at the flow front of the dilute part influences the dynamics of PDCs as a whole.
The results in Fig. 12 are preliminary, and a comprehensive understanding of the dynamics of PDCs should consider many other effects, such as the expansion of entrained air due to heating by pyroclasts, density stratification inside the overlying dilute layer, diffusion of the pore pressure and entrainment of air in the underlying dense layer, and the transport of particles from the underlying dense layer to the overlying dilute layer (e.g., Andrews 2014; Breard and Lube 2017; Bursik and Woods 1996; Dufek and Bergantz 2007; Esposti Ongaro et al. 2016; Ishimine 2005; Roche et al. 2008; Wilson and Walker 1982). Nevertheless, preliminary results (not shown here) have already indicated the diversity of the interplay between the dilute and dense layers, which depends on the initial particle concentration and grain size. The interaction also influences the sedimentation process from the PDCs (Fujii and Nakada 1999). A systematic parametric study of the twolayer PDC model using the BC model is in progress, with the aim of accounting for the diversity of PDC deposits.
Conclusion
A numerical shallowwater model of simulating gravity currents for a wide range of ρ _{c}/ρ _{a} has been proposed. In the model, the effects of varying ρ _{c}/ρ _{a} are taken into account via the front condition. We have assessed two types of numerical models for the front condition (the Boundary Condition (BC) model and the Artificial Bed (AB) model) by comparing their numerical results with the analytical results. The results from the BC model agree well with the analytical results when ρ _{c}/ρ _{a}≲10^{2}. In contrast, the AB model generates good approximations of the analytical results for \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\). On the basis of these results, we have developed a twolayer model of pyroclastic density currents (PDCs), in which the BC model is used for the overlying dilute part (ρ _{c}/ρ _{a}=10^{0}−−10^{1}) and the AB model is used for the underlying dense part (ρ _{c}/ρ _{a}=10^{2}−−10^{3}). This twolayer model successfully simulates some essential features of PDCs with strong density stratification.
Appendix: twolayer model of pyroclastic density currents
Here, ϕ(x ^{∗},t ^{∗}) is volumetric concentration, and the subscripts L,H,s,g, and a denote the dilute (i.e., lowparticle concentration) and dense (i.e., highparticle concentration) layers, solid particles, volcanic gas, and air, respectively. W _{s} is the settling velocity of the particles from the base of the dilute layer, E is the entrainment coefficient, \(\tau _{\mathrm {m}}^{*}\) is the interfacial shear drag, and \(\tau _{\mathrm {b}}^{*}\) is basal resistance of the dense layer.
In the dilute layer, it is assumed that turbulent mixing is sufficiently intense to maintain vertically uniform volumetric concentrations (e.g., Bonnecaze et al. 1993; Bursik and Woods 1996; Johnson and Hogg 2013). The dense layer is assumed to have a constant bulk density ρ _{H}=ρ _{s} ϕ _{sH}+ρ _{g}(1−ϕ _{sH}), where the particle volumetric concentration ϕ _{sH} is set to 0.4 (Breard et al. 2016).
Interactions between the two layers are treated in the source terms of Eqs. (21)–(26) (i.e., the righthand sides of the equations). Particle settling from the dilute layer to the dense layer is taken into account in the second source terms in Eqs. (21) and (22), the source terms in Eqs. (23) and (25), and the third source term in Eq. (24). The acceleration of the dilute layer over the basal contact is taken into account in the first source term in Eq. (22). The pressure gradient on the dense layer exerted by variations in the height of the dilute layer is taken into account in the fourth source term in Eq. (24).
The entrainment of ambient air into the dilute layer is taken into account in the first source term in Eq. (21) and the source term in Eq. (26). Thermal expansion of the entrained air is neglected here for the sake of ease. Air entrainment is also assumed to occur on the upper surface of the dilute layer (e.g., Bursik and Woods 1996; Johnson and Hogg 2013), although a different process for entrainment was recently proposed by Sher and Woods (2015). We adopted the entrainment coefficient proposed by Johnson and Hogg (2013): i.e., E=0.075/(1+27Ri), where \(Ri\equiv \left (\rho _{\mathrm {L}}^{*}\rho _{\mathrm {a}}\right)gh_{\mathrm {L}}^{*}/\left (\rho _{\mathrm {L}}^{*}u_{\mathrm {L}}^{*2}\right)\).
(Figure 11 b; see Roche et al. 2008 for details), where the dynamic basal friction angle δ is set to 20° (Doyle et al. 2008).
which is numerically treated by the BC model. The front condition of the dense layer, given by Eq. (3), is numerically treated by the AB model with ε=10^{−10}.
In calculating the twolayer PDC model (Fig. 12), the conservation Eqs. (21)–(26) were numerically solved. In calculating the onelayer dilute PDC model (Fig. 10), we solved conservation Eqs. (21), (22), (25), and (26), where \(h_{\mathrm {H}}^{*}\approx 0\) and \(u_{\mathrm {H}}^{*}\approx 0\). In calculating the onelayer dense PDC model (Fig. 11), we solved conservation Eqs. (23) and (24). For numerical simulations, we used a fractionalstep method to solve the conservation equations with source terms (e.g., LeVeque 2002), and the HLL approximate Riemann solver to calculate the intercell flux of the equations (e.g., Toro 2009).
Abbreviations
 AB model:

Artificial Bed model
 BC model:

Boundary Condition model
 PDC:

Pyroclastic density current
Declarations
Acknowledgements
We thank Editor Colin J. N. Wilson, Reviewer Eric C. P. Breard, and an anonymous reviewer for their thoughtful comments.
Funding
Not applicable.
Authors’ contributions
HAS carried out this study under the supervision of TK and YJS and prepared the first draft of the manuscript. All authors contributed to writing the manuscript and approved the final version.
Authors’ information
HAS is a Ph.D. candidate supervised by TK and YJS. TK is a Professor at the Earthquake Research Institute, the University of Tokyo. YJS is an Assistant Professor at the Earthquake Research Institute, the University of Tokyo.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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