GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-11-2923-2018faSavageHutterFOAM 1.0: depth-integrated simulation of dense snow avalanches on natural terrain with OpenFOAMOpenFOAM for dense snow
avalanchesRauterMatthiasmatthias.rauter@uibk.ac.athttps://orcid.org/0000-0001-7829-6751KoflerAndreasHuberAndreasFellinWolfgangDivision of Geotechnical and Tunnel Engineering, Institute of
Infrastructure, University of Innsbruck, Innsbruck, AustriaDepartment of Natural Hazards, Austrian Research Centre for Forests
(BFW), Innsbruck, AustriaNorwegian Geotechnical Institute, Oslo,
NorwayDivision of Hydraulic Engineering, Institute of
Infrastructure, University of Innsbruck, Innsbruck, AustriaMatthias Rauter (matthias.rauter@uibk.ac.at)23July2018117292329396March201812March201813June20182July2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://gmd.copernicus.org/articles/11/2923/2018/gmd-11-2923-2018.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/11/2923/2018/gmd-11-2923-2018.pdf
Numerical models for dense snow avalanches have become central to hazard zone mapping
and mitigation. Several commercial and free applications, which are used on a
regular basis, implement such models. In this study we present a tool based
on the open-source toolkit OpenFOAM® as an
alternative to the established solutions. The proposed tool implements a
depth-integrated shallow flow model in accordance with current practice. The
solver combines advantages of the extensive OpenFOAM infrastructure with
popular models from the avalanche community. OpenFOAM allows assembling
custom physical models with built-in primitives and implements the numerical
solution at a high level. OpenFOAM supports an extendable solver structure,
making the tool well-suited for future developments and rapid prototyping. We
introduce the basic solver, implementing an incompressible, single-phase
model for natural terrain, including entrainment. The respective workflow,
consisting of meshing, pre-processing, numerical solution and
post-processing, is presented. We demonstrate data transfer from and to a
geographic information system (GIS) to allow a simple application in
practice. The tool chain is based entirely on open-source applications and
libraries and can be easily customised and extended. Simulation results for a
well-documented avalanche event are presented and compared to previous
numerical studies and historical data.
Introduction
Numerical avalanche modelling has become an important and well-accepted
ingredient in hazard zone mapping. All popular tools rely on depth-integrated
flow models and only a few academic exceptions
are known (; ;
;
). Depth-integrated flow models, widely known as
shallow water equations, have a long tradition in hydraulic modelling
e.g., dating back to
. This approach is commonly applied in academia and
in practice because it reduces the computational effort to a level at which physical simulations of realistic flows are feasible. The first application
to gravitational mass flows is attributed to and the
first formal derivation and analysis of the underlying model to
. Since then, the mechanical model
has been continuously improved and extended to, for example, simple, two-dimensional
surfaces , complex, shallow surfaces
, or curved and twisted flow paths
. Finally, respective
models have been adapted to natural, i.e. arbitrary but mildly curved, terrain making simulations of real case avalanches possible. The limitation
to mildly curved terrain requires the flow thickness to be small in relation
to the curvature radius of the surface.
proposed a model embedded in an ordinary Cartesian coordinate system as an
alternative to the complex curvilinear coordinate system used by
.
, , and
recently follow a similar approach.
apply a non-orthogonal local coordinate system
but without incorporating the respective
correction terms . A Lagrangian solution, which
has some advantages for natural terrain, has been presented by
and later on by and
.
Beside improvement of the underlying mechanical model, various physical
processes have been added to governing equations, such as multiple phases
e.g., entrainment e.g.,
improved basal friction relations
e.g.,
for a review, see, compressibility
e.g. or thermodynamic
processes e.g..
In this work, we strictly distinguish between a mechanical model and process
models. The mechanical model consists of basic conservation equations and
their reformulation, e.g. in terms of depth integration. Process models, on
the other hand, describe the closure of governing equations, for example with constitutive models. The combination of the mechanical model and all
closures is called flow model or physical model throughout this work.
There are several numerical methods to solve the respective mathematical
equations. Basically, most methods can be classified as finite-difference
methods e.g., finite-element methods
e.g., finite-volume methods
e.g. or as Lagrangian particle methods
e.g.. Specialised differencing schemes (e.g.
upwind, TVD, NVD) prevent oscillations e.g..
Shallow granular flow models have been carefully validated over the last few
decades. This includes back-calculations of small-scale experiments
for a review, see, large-scale experiments
e.g., historic snow avalanches
e.g. and rock avalanches
e.g.. Shallow flow models have various
weaknesses, such as the limitation to mildly curved terrain or the missing
resolution in surface-normal direction. However, they have proven to be a
good trade-off between accuracy and computing time and thus useful for many
applications.
Shallow flow models gained popularity through commercial software packages:
DAN , SamosAT , FLATModel
and RAMMS implement
such models and are used regularly in practice. Open-source alternatives
include TITAN2D , r.avaflow
and an extension to the
CFD toolkit (computation fluid dynamics) GERRIS
. From an academic viewpoint, open-source
applications have various advantages over their commercial counterparts;
for example, users can view and modify the source code to gain a better understanding
of the software and adapt the flow model without re-implementing basic models
and numerical methods from scratch.
Geographic information systems (GISs) are commonly applied in hazard zone
mapping. Therefore numerical simulation tools are usually incorporated or
linked to these systems to streamline the respective workflow. GIS allows
user-friendly data input, post-processing and the production of publication-quality maps.
Recently, proposed a shallow granular flow model,
expressed in terms of surface partial differential equations
and presented an
open-source implementation based on the CFD toolkit
OpenFOAM®. The
underlying mechanical model is widely similar to the classic
model and its derivations.
One particular advantage of an OpenFOAM solver is the well-designed,
object-oriented source code. This makes the code cleaner than comparable
solutions as it hides implementation details, such as numerical schemes,
input/output or
inter-process communication, behind well-defined interfaces. The top-level
solver mimics the tensorial notation of partial differential equations, and
specific implementations of, for example, interpolation schemes, are
exchangeable without changing the top-level source code. This enables the
separation of physical models and numerical solution, which allows a
streamlined interdisciplinary development process. Process models, e.g. of
entrainment and basal friction, can be incorporated similarly, keeping the
source code clean and easy to extend.
The OpenFOAM solver, presented here, implements an incompressible
single-phase model including various basal friction and entrainment closures.
The solver is called faSavageHutterFoam, indicating that the
underlying mechanical model is similar to the one of
but with exchangeable
closure models. This model is, to some extent, suitable for dense snow
avalanches and constitutes the baseline for complex flow models, as employed
by, for example, or .
Moreover, the underlying method has been developed to simplify coupling with
three-dimensional ambient flows
,
which enables the development of models for mixed snow avalanches
e.g. and turbidity currents
e.g..
The purpose of this paper is to present the capability of the new OpenFOAM
solver and the model. The solver is evaluated and
validated for snow avalanches on natural terrain. We present the basic flow
model, as well as methods and tools to incorporate natural terrain and GIS
data in OpenFOAM simulations. Also, the export of OpenFOAM results to a GIS
for post-processing and visualisation is demonstrated. Results for a
well-documented avalanche event are presented and compared to historical
records and results of SamosAT. All underlying source code (except SamosAT)
and data are available free of charge to encourage reproduction, improvement
and cross-validation.
MethodFlow model
Historically, shallow granular flow models have been set up in
surface-aligned, curvilinear coordinates, leading to a two-dimensional system
of partial differential equations
e.g..
follow a different approach see
also and
formulate the mechanical model in terms of surface partial differential
equations SPDEs; e.g.. Respective SPDEs
are defined on a surface Γ, embedded in three-dimensional space, which
represents the mountain topography. This approach, popular in the thin
liquid-film community e.g., avoids
transformations into the surface-aligned coordinate system and thus complex
metric tensors. Considering the relative shallowness of the avalanche, it can
be treated as a thin layer flowing along the mountain surface. The governing
equations describe the motion of the avalanche in three-dimensional space
along this surface. Consequently, velocity is a three-dimensional vector
field and contains all information on flow direction and respective effects,
such as centrifugal forces. Resulting SPDEs can be solved with various
methods, e.g. the finite-element method e.g.
or the finite-area method, a modified finite-volume method
.
Definition of velocity u, flow thickness h and basal
pressure pb on a control volume. A hydrostatic and linear
pressure distribution is assumed. The shape of the velocity profile is
commonly ignored in governing equations . Flow thickness
h is measured normal to the basal surface Γ. The curvature radius of
the surface Γ is assumed to be much bigger than the flow thickness.
Mechanical model
A basic shallow granular flow
model can be written in terms of surface partial differential equations
as
Multiplications between vectors represent the outer product
uv=u⊗v.
Equations () to () are equivalent to a
Savage–Hutter-like system, consistently extended to complex but mildly
curved terrain and entrainment. The notation as SPDE makes extension to
complex terrain straightforward and implementation into SPDE environments,
e.g. OpenFOAM, possible. A formal derivation is given by
. Here, we aim to deliver a short and descriptive
introduction.
Equation () represents the depth-integrated continuity equation,
Eq. () the surface-tangential momentum conservation equation
and Eq. () its surface-normal counterpart, defined at all
points xb on the surface Γ⊂R3,
representing the mountain surface. The time is denoted as t. The unknown
fields are the surface-normal flow thickness h(xb) (see
Fig. ), the depth-averaged flow velocity
u‾(xb)∈R3, defined as
u‾(xb)=1h(xb)∫0h(xb)u(xb-nbz′)dz′,
and the basal pressure pb(xb). The density
ρ is assumed to be constant. Note that the earth pressure theory
e.g. has been replaced with
the hydrostatic pressure assumption, as in most practical applications
e.g.. Moreover, Eqs. () to
() are written in conservative form. Therefore, there is
no entrainment term in Eq. (), which would show up in a
non-conservative formulation. The first terms in Eqs. ()
and () represent the temporal derivative, i.e. the local
change in mass and momentum, respectively. The second terms in
Eqs. () and () are the respective advection
terms. The right-hand side of Eq. () represents mass growth due
to entrainment. The first, second and third terms on the right-hand side of
Eq. () represent surface-tangential components of basal
friction, gravitational acceleration and lateral pressure gradient,
respectively. The surface-normal components of these terms appear in the
surface-normal momentum conservation equation (Eq. ). This
equation is used to calculate the basal pressure, represented by the last
term.
In the framework of SPDEs, the normal vector field
nb(xb)∈R3 of the surface
Γ is sufficient to describe all major curvature effects. This is
realised by calculating all contributions to conservation equations in the
global coordinate system and projecting results onto the surface and the
surface-normal vector. These projections are explained in detail in
Appendix . Surface-tangential and normal components
contribute to local acceleration and basal pressure, respectively. This
follows from the assumption that movement is constrained in surface-normal
direction, which is enforced by a mechanical force, namely the basal
pressure. The gravitational acceleration g, for example, is split
into a surface-tangential component,
gs=I-nbnb⋅g,
and a surface-normal component,
gn=nbnb⋅g.
The gradient operator ∇ denotes the three-dimensional derivative
along the surface . If the responding result
is a three-dimensional vector field (e.g. gradient of a scalar field or
divergence of a tensor field), it can be split, similar to the gravitational
acceleration, into a surface-tangential component,
∇s=I-nbnb⋅∇,
and a surface-normal component,
∇n=nbnb⋅∇.
For simply curved surfaces, the given relation matches the model of
, as shown by .
Process models
There are various user-selectable models, describing basal friction
τb(xb) and entrainment rate
q˙(xb), to close the system of equations. To
reassemble the traditional model often called Voellmy or Voellmy–Salm
model;, as applied, for example, by
, the basal friction is described following
,
τb=μpbu‾|u‾|+u0+ρgξ|u‾|u‾.
Therein, μ and ξ are constant parameters, although they may depend on
avalanche size and surface roughness or flow
regime . The small value u0
(10-7ms-1 here) avoids divisions by zero and regularises
the relation near standstill, where the original function is discontinuous.
This regularisation, combined with the employed time integration scheme
implicit three-level second-order;,
leads to well-defined behaviour in the runout zone, where the velocity is
nearly zero . This allows the avalanche to reach very
low velocities in the runout zone, which are lower than the tolerance of the
solver and thus virtually zero. For characteristic avalanche velocities, i.e.
|u‾|>100u0, this value has no relevant effect on the
dynamic behaviour. Previously, this issue has been addressed with operator
splitting and explicit stress reduction e.g., which is not required in the proposed
scheme.
The entrainment rate is calculated, based on an empirical erosive entrainment
model, as
q˙=τb⋅u‾ebforhmsc>0,0forhmsc=0,
where eb is the specific erosion energy
. Entrainment is restricted by the available
mountain snow cover thickness hmsc. The initial mountain snow
cover thickness is calculated following ,
using a linear approach,
hmsc(z)=Hmsc(z0)+∂Hmsc∂zz-z0cos(ζ),
where z is the surface elevation (corresponding to the vertical coordinate
in the numerical model) and z0 is the elevation of a reference station,
which has to be provided by the user, alongside with the base value
Hmsc(z0) and the growth rate ∂Hmsc∂z. ζ is the angle between the
gravitational acceleration and the surface-normal vector. Its further
evolution is described by the conservation equation
∂hmsc∂t=-q˙ρ.
Undershoots, i.e. hmsc<0, are prevented with a regularisation
similar to Eq. (). This can be realised by multiplying the
entrainment rate q˙ with
hmschmsc+h0, where h0 is a small
value, similar to u0.
Numerical solution
The governing equations are solved with an implicit, conservative,
finite-area method , using the respective OpenFOAM
library . The finite-area method is similar to the
well-known finite-volume method e.g. but with
appropriate differential operators for SPDEs: Eqs. ()
and (). We apply first- (upwind scheme) and second-order
accurate spatial differencing schemes. First-order schemes converge more
slowly in terms of mesh refinement due to their high numerical diffusivity.
However, they effectively prevent oscillations and increase the stability of
the solver. Oscillations in second-order accurate simulations are prevented
with a normalised variable diagram (NVD) scheme for unstructured meshes,
known as the Gamma scheme . NVD schemes blend upwind and
a higher-order scheme to combine advantages of both methods.
As mentioned before, OpenFOAM utilises capabilities of C++ to make top-level
source code appear similar to the tensor notation of partial differential
equations. The conservation equation (Eq. ), for example, can be solved
with the following lines of code using OpenFOAM:
phis is the velocity edge field seefor
details, and dqdt is the source term
incorporating entrainment. Momentum conservation equations
(Eqs. and ) look similar
see, and conservation equations for arbitrary
fields e.g. random kinetic energy, can
be added with the same syntax.
Simulation evaluation
We use an established implementation of the same flow model, SamosAT (version
2017_07_05) , for comparison.
The main difference between SamosAT and the presented OpenFOAM solver is the
solution method. SamosAT solves similar governing equations, slightly adapted
to fit into the respective framework, with smoothed-particle hydrodynamics
(SPH). This approach follows a Lagrangian description, making the handling of
complex terrain simpler . Therefore, SamosAT
provides an excellent reference to validate avalanche models for complex
terrain. The second term on the right-hand side of Eq. ()
was deactivated in OpenFOAM computations to reassemble the mechanical model
as implemented in SamosAT. This term is usually small and can be safely
neglected . However, it is shown in equations to
preserve the similarity between Eqs. ()
and ().
We compare simulations using the 1 kPa isoline of the dynamic peak
pressure, defined as
pdyn(xb)=maxtρ|u‾(xb,t)|2.
Definitions of hazard zones are based on this threshold in many European
countries and are therefore often used for the
evaluation of relevant models
e.g..
In addition to the comparison with a reference implementation, we present a
comparison with historical records from a catastrophic event. A common method
to document avalanches is the delineation of deposition. This information is
also available for the presented case study. Deposition processes are not
explicitly included in the flow model due to depth integration. However, the
general form and size of the deposition should be reproduced by the model to
be useful for hazard zone mapping. This is problematic in some
implementations, e.g. SamosAT, due to missing regularisation of the friction
term, but it is possible with the proposed method.
We apply model parameters (μ, ξ, eb) optimised for
SamosAT , and the comparison is conducted on a
qualitative level.
Finally, we evaluate OpenFOAM simulations with regard to convergence during
mesh refinement to give a quantitative estimation of numerical uncertainties
as recommended by . The numerical solution
should converge to the unknown analytical solution with increasing grid
resolution, and the numerical uncertainty should decay with the order of the
applied method. Richardson extrapolation allows us to estimate the numerical
uncertainty, using results of three different meshes. This way, the expected
convergence can be verified and the numerical uncertainty quantified.
Simulation set-up
The precondition to conduct simulations in OpenFOAM is a mesh, describing the
geometry of the problem. For SPDEs, e.g. shallow flow models, a surface mesh,
matching the slope topography, is sufficient and no volume mesh is required.
In practice, however, three-dimensional meshing tools can be used to create a
volume mesh, the boundary of which can be used as surface mesh.
Simulation set-up and tool chain. The tool chain consists
exclusively of open-source applications. Individual applications and process
models can be replaced with custom ones. Parameters for python scripts are
provided via a command line interface. Parameters for OpenFOAM applications
are provided through OpenFOAM dictionaries. Domain decomposition and
reconstruction, which is handled by separate applications, is not shown.
OpenFOAM reads initial conditions from the folder “0” and writes results to
folders named after the corresponding time step (“1”, “2”, etc.). Details
on OpenFOAM formats can be found in .
Topography is usually available as a digital elevation model (DEM) in GIS
formats, yielding elevation on a regular two-dimensional grid. The relevant
part of the topography is re-sampled with cubic splines, triangulated and
stored as an STL file e.g. to prepare it for
meshing. We chose the meshing application cfMesh
because of its good integration in OpenFOAM and its clean boundary meshes.
cfMesh requires a closed triangulated surface to create a volume mesh. This
is the case for all general purpose meshing tools, and cfMesh can be replaced
easily in our tool chain, for example with Netgen seefor an application. Various other meshing
tools can be applied and OpenFOAM provides a large range of mesh conversion
tools. The closed surface can be assembled from a triangulation of the
mountain surface, sidewalls and the respective top boundary. The resulting
surface and volume mesh are presented in Fig. b and c.
Refinement near the mountain surface reduces the amount of required volume
cells, while keeping the number of surface cells high. The resulting mesh is
also valid for three-dimensional simulations with, for
example, Navier–Stokes equations, as conducted by, for
example, , ,
, ,
and .
The boundary mesh, describing the mountain surface, is shown in
Fig. d. The shallow flow model is solved on this
surface mesh. We used polygonal-dominated (or volumetric
polyhedral-dominated) meshes for simulations for stability and accuracy
reasons . Triangular (or volumetric
tetrahedral) meshes have been evaluated as well. However, second-order
accurate simulations on triangular meshes failed, while first-order accurate
simulations are virtually identical to the respective simulations on
polygonal-dominated meshes.
The release area, acting as an initial condition, is provided as a polygon in
an ESRI shape file format . To find all surface
cells within the given polygon, the algorithm as
implemented in OpenFOAM is applied. The mountain snow cover
hmsc of the corresponding cells is then transferred to the flow
thickness h to create a suitable initial condition. The release area for
our case study, taken from , is shown in
Fig. a as a polygon and as a set of surface cells in
Fig. d.
Meshing tool chain: the terrain data are usually available as raster
data (a). Triangulation of the relevant area and adding walls and a
top boundary yields a closed triangulated surface (b; sharp edges
are highlighted black). This surface can be processed by most meshing tools;
here, we apply cfMesh to get a polyhedral-dominated finite-volume
mesh (c). The bottom boundary surface of the finite-volume mesh
builds the foundation for the finite-area mesh used for
simulations (d). Note that we show a very coarse mesh for the sake
of visibility of the edges. Terrain data: Amt der Tiroler Landesregierung
(AdTLR). EPSG coordinate reference system: 31254.
Time series of an OpenFOAM simulation with mean cell size Δ=7.45 m and first-order interpolations in ParaView. The colour scale
represents flow thickness, which is clipped at 0.5 m. Terrain data:
AdTLR.
The solver reads the surface mesh and initial conditions, as well as physical
models, numerical schemes and constants to initialise the simulation (see
Fig. ). The respective entries can be found in the
designated locations, according to the usual practice in OpenFOAM
. The solver can run on multiple processors using
domain decomposition and message passing
interface (MPI).
User-defined friction and entrainment models can be loaded at run-time,
meaning that the user does not have to recompile the solver to add a custom
friction or entrainment model. The same is the case for general purpose
functions which are triggered at the end of every time step. Here, we used
this interface to calculate and record the dynamic peak pressure at run-time,
without the necessity to save multiple time steps or to change solver source
code. Similar functions can be used to check mass, momentum or energy
conservation, record specific data (e.g. time line at a certain point), or to
manipulate fields during run-time, e.g. to trigger secondary slabs.
Simulation results are written to hard disk in the usual OpenFOAM file format
for post-processing, evaluation and simulation
restart. The simulation set-up, all involved applications, and all
intermediate and final files are presented in Fig. . The
tool chain is modularly assembled from various open-source applications.
Single modules, such as mesher, solver or friction model, can be replaced
easily.
Post-processing and visualisation
Post-processing and visualisation of OpenFOAM simulations is commonly
performed using ParaView® (see
Figs. , and ).
ParaView is an open-source data analysis and visualisation application. It
can read and visualise OpenFOAM files, and they can be used for further
operations, such as the calculation of contour lines. To integrate GIS
applications in post-processing, results can be exported to common GIS file
formats. Contour lines can be exported to ESRI shape file format with a
custom python extension based on the library pyshp
(Figs. c, and ).
Alternatively, individual cells and respective field values can be exported
as polygons (Fig. a) or points
(Fig. b) to ESRI shape files.
Perspective view on the OpenFOAM simulation with mean cell size
Δ=7.45 m and first-order interpolations in ParaView. The colour
scale represents flow thickness, which is clipped at 0.5 m. Terrain data:
AdTLR.
The flow thickness field h at time t=40 s for a simulation
with mean cell size Δ=7.45 m; first-order interpolations. The
figure shows four methods to export and analyse results in GIS: export of
cells as polygons (a); export of cell centres as
points (b); export of contour lines as polygons (c);
remapping of the unstructured finite-area mesh to a regular
raster (d). The raster has been created by converting point data to
a raster file in QGIS. The resolution of the DEM is 10 m, results have
been mapped to a 5 m grid. Terrain data: AdTLR.
To generate regular raster files, the unstructured OpenFOAM mesh and
associated fields have to be mapped to a structured Cartesian grid
(Figs. d and ). These and other
approaches allow an almost seamless integration into general purpose GIS
applications, as shown in the following case study. Here, we utilise
foam-extend-4.0 with a custom solver, python 2.7.12 with numpy 1.11.0, scipy
0.17.0 and pyshp 1.2.3 for shape file export, ParaView 5.0.1, and QGis 2.8.6.
Case study
In this work we focus on the Wolfsgruben avalanche. The event from
13 March 1988, when the avalanche struck inhabited areas, has been repeatedly
used as benchmark for avalanche simulations, most recently by
. We chose this example because the relevant
data are freely available, making reproduction and cross-validation possible.
The mountain snow cover thickness for the specific event can be described
with the parameters Hmsc(z0)=1.61 m, z0=1289 m and
∂Hmsc∂z=8×10-4. Physical
parameters to reassemble the runout properly are μ=0.26, ξ=8650ms-2 and eb=11500Jkg-1.
These parameters were optimised in a previous study using SamosAT
.
Numerical parameters for OpenFOAM see have been
chosen such that they do not influence the results, while keeping the solver
as stable as possible. The appropriate mesh resolution for OpenFOAM has been
identified using a mesh refinement study, which is presented alongside the
results. The simulation duration has been set to 150 s. This duration is
sufficient to reach standstill (i.e. a velocity lower than the solver
tolerance, |u‾|<10-5ms-1) in the runout
zone and thus virtually unchanging deposition. We decomposed the simulation
domain into four parts for OpenFOAM and all simulations have been conducted
on a Quadcore Intel Core i7-7700K @ 4.20 GHz and 32 GB DDR4 Ram @
2.667 GHz.
SamosAT utilises a grid with 5 m resolution, and we follow recommendations
in terms of appropriate particle numbers and other numerical parameters. The
interpolation method has been varied between interpolation on a grid
(SPH-mode 0) and interpolation on particles (SPH-mode 1) to get an insight
into the numerical uncertainty.
ParaView renderings are presented in Fig. for multiple
time steps, showing the dynamic behaviour of the avalanche. A perspective
ParaView rendering is shown in Fig. . The avalanche
follows the narrow channel directly beneath the release area. Small portions
of the avalanche overflow the left and right humps in some simulations, which
can be seen in the peak dynamic pressure
(Fig. ).
The results at time step t=40 s have been exported to QGIS using various
methods; see Fig. . Affected areas (i.e. 1 kPa
isolines), as predicted by OpenFOAM and SamosAT, are shown in
Fig. . Variations due to different interpolation schemes
are shown for both implementations to give an insight into the numerical
uncertainty.
The influence of the mesh resolution on the affected area is shown in
Fig. for the OpenFOAM solver. Respective mean cell
sizes, an estimation of the numerical uncertainty following
and execution times (excluding time for mesh
generation, which may take several minutes) are presented in
Table . Here, the runout is defined as the length of the
central avalanche path (see Fig. ) within the affected
area. The central avalanche path has been taken from
. The mean cell size is defined as the square
root of the mean cell area. For comparison, execution times for SamosAT are
98 s (SPH-mode 0) and 368 s (SPH-mode 1). One should keep in mind that
SamosAT only utilises a single processor core while OpenFOAM utilises all
available cores. Moreover, execution times should be seen as rough estimates
because they depend on various factors, such as the number of saved time
steps, debug messages and compile options.
Mesh size, runout, error estimation and execution time for different
OpenFOAM simulations. Base cell size and refinements refer to parameters of
cfMesh.
Deposition (i.e. flow thickness field h in the last time step) of the
OpenFOAM solution is shown in Fig. alongside with the
documentation.
Comparison of OpenFOAM first order (blue, dashed), OpenFOAM second
order (blue), SamosAT SPH 0 (red, dashed) and SamosAT SPH 1 (red) in terms of
1 kPa isolines (affected area). OpenFOAM results are based on the mesh
with cell size Δ=7.45 m. The documented release area (orange area)
and documented deposition area (blue area) are shown for orientation. The
shape and reach of the main avalanche branch are similar in all simulations;
secondary branches differ to some extent. Overview (a) and focus on
the runout zone (b). Terrain data: AdTLR.
Mesh refinement and convergence study for the OpenFOAM solver. Four
mesh sizes and both interpolation schemes, first-order upwind (dashed line)
and second-order Gamma (solid line) have been evaluated. The central
avalanche path from is shown in black.
Terrain data: AdTLR.
Flow thickness field h at t=150 s of the second-order OpenFOAM
simulation (Δ=3.75 m) and the documented deposition area. The flow
thickness field in the last time step should roughly replicate the
deposition. The bulge on the orographic right side of the deposition area is
not matched by any simulation. However, some interesting details, such as the
tail of the avalanche are represented well in OpenFOAM simulations. Map data:
http://basemap.at (last access: 1 March 2018).
Discussion and conclusion
Results of the new OpenFOAM solver are very similar to SamosAT. Differences
between SamosAT and OpenFOAM are in the range of numerical uncertainty, and
differences between interpolation methods are of a comparable size. This
uncertainty has to be expected; in fact, it is well known in the CFD
community that numerical schemes and implementation details influence results
if they are not converged to the analytical solution
e.g.. In the case of gravitational mass
flows, numerical uncertainty plays a minor role, since underlying models,
parameters, terrain and snow cover data are affected by substantially higher
uncertainty. This is shown by a comparison of the documented deposition with
the result of an OpenFOAM simulation in Fig. . Although
parameters have been optimised to the specific event, all simulations differ
significantly from documentation. In particular, the large bulge on the
orographic right side of the deposition area is not matched by any
simulation. However, some details, such as the form of the tail and the
position where the deposition expands, are accurately simulated by the
OpenFOAM solver. Significant differences between simulation and documentation
are not limited to the presented case and have been observed before, for
example, by . We deduce that numerical errors are much
smaller than the expected model error. Under these circumstances, a
quantitative comparison between implementations as, for example,
byfor basal friction models is not appropriate.
The refinement study shows that in the presented case, the simulated runout
reduces with increasing mesh refinement (Fig. ).
Simulations on fine meshes are stopped by the first embankment, simulations
on coarser grids overflow it and reach the next embankment. This is
reasonable, considering the higher diffusivity and lower curvature of coarser
meshes. However, this trend should not be taken for granted for other cases
and a refinement study should always be conducted to get an insight into the
numerical uncertainty. Results indicate that a cell size of approximately
3.75 m is required in OpenFOAM to achieve convergence with respect to
practical applications. The numerical uncertainty cannot be calculated for
the coarsest and finest mesh, since three simulations are required to conduct
a Richardson extrapolation. It has to be noted that all simulations are based
on the same DEM with a grid size of 10 m. The influence of terrain model
quality see, e.g., on simulation results is
not investigated.
The execution time of the OpenFOAM solver is acceptable for coarse meshes but
increases with the square of the number of cells because the time step
duration has to be reduced similarly to cell size. The OpenFOAM solver is
noticeably slower than SamosAT, especially when considering OpenFOAM's
multiprocessing capabilities. For applications where fast execution is
imperative, such as parameter studies, SamosAT may be the appropriate choice.
There is potential for future optimisation in OpenFOAM; in particular, the
implicit time integration scheme is expensive and should be replaced with a
simpler explicit one. However, the implicit solution strategy, in combination
with the regularised friction relation, leads to satisfying behaviour in the
runout zone. In contrast, the simple explicit solution strategy, for example
from SamosAT, leads to a continuous creeping of the deposition, meaning that
the final flow thickness cannot be compared with the deposition, as noted by
and .
The stability of the OpenFOAM solver is strongly influenced by mesh quality.
Simulations with polygonal-dominated surface meshes showed an acceptable
stability for first- and second-order interpolations. The high influence of
the three-dimensional mesh on stability and its computationally expensive
creation is the main drawback of the proposed method. This is, however, also
a big advantage, allowing simple coupling with three-dimensional ambient
flows, as conducted by .
Summary and outlook
This paper shows the application of a finite-area scheme for shallow granular
flows to snow avalanches on natural terrain.
Specific processes, such as entrainment, have been added to the basic model
to replicate the traditional model as implemented in SamosAT
.
Various simulations with the new OpenFOAM solver have been conducted. Methods
and tools to incorporate the OpenFOAM solver in GIS have been presented.
These tools allow the integration of OpenFOAM in hazard mapping workflows and
thus the validation of the OpenFOAM solver with a reference implementation,
herein SamosAT.
The application of three-dimensional Cartesian coordinates allows simple
coupling with GIS applications because no coordinate transformations are
required. Unstructured meshes, on the other hand, require re-sampling to
structured meshes or data transfer in the form of polygons. This incorporates
an additional effort compared to simulations on structured meshes, as
conducted for example by .
The OpenFOAM solver roughly reproduces the results of SamosAT. Differences
are within the expected numerical uncertainty. A comparison of numerical
results to a documented event suggests that model uncertainty is
substantially higher than numerical uncertainties.
The major advantage of OpenFOAM is the object-oriented open-source code,
which can be extended easily. The flexible code structure allows fast
application of new models to real case examples. This especially qualifies
the proposed method for model development and academic purposes. Moreover,
the vast majority of source code is shared within the OpenFOAM community,
leading to faster development of core features and higher code quality.
The finite-area scheme allows a description in terms of surface partial
differential equations , which leads to
simple and expressive governing equations. However, this comes at the cost of
a complex three-dimensional surface mesh. The projection of the governing
equations on a plane surface following, for example,
may be beneficial for some applications. The
three-dimensional surface mesh can also be an advantage, allowing a simple
coupling with three-dimensional ambient two-phase models for powder clouds
. The presented meshing method, creating a
finite-volume and the corresponding finite-area mesh, is viable for such
simulations as well.
Future steps will incorporate the
optimisation of the solver in terms of stability and execution time. Mesh
generation and the integration of geographic information systems will be
further streamlined. The limitation to mildly curved terrain should be
eliminated, as this assumption is violated in many practical cases. We aim to
implement more complex models, suitable for mixed snow avalanches
e.g. and debris flow
e.g. in the near future.
Coupling of the dense flow model proposed here with three-dimensional
two-phase models for the powder cloud regime e.g. is planned as well.
The OpenFOAM solver, core utilities and the case study
presented are available in the OpenFOAM community repository
(https://develop.openfoam.com/Community/avalanche, last access: 1 March 2018)
and integrated as a module within OpenFOAM-v1712.
The complete code (based on foam-extend-4.0), including python scripts for
GIS integration and the simulation set-up including the underlying raw data,
is included in the Supplement and available at
https://bitbucket.org/matti2/fasavagehutterfoam (last access: 1 March 2018).
Understanding projections in surface partial differential equations
Here we briefly explain the concept of projections within the framework of
surface partial differential equations. These projections are widely used in
computational fluid dynamics, usually when surfaces in three-dimensional
space are considered. We do not focus on mathematical formalities and this
section cannot replace the formal derivation of . We
want to emphasise that no surface-aligned coordinate system is required
throughout the whole process, and the reader is encouraged to adhere to
global Cartesian coordinates. For simplicity we present a discretised
finite-area cell, which has been extruded by flow thickness h to present
the flowing mass; see Fig. .
Splitting gravitational acceleration into a surface-tangential and
surface-normal part with simple projections to the surface-normal vector
nb.
We begin by splitting a simple vectorial entity, the gravitational
acceleration g∈R3, into a surface-normal component,
gn∈R3, and a surface-tangential component,
gs∈R3, as shown in
Fig. . The magnitude of the surface-normal component
can be calculated using the scalar product and the surface-normal vector:
‖gn‖=nb⋅g,
which corresponds to a projection of g on nb. The
surface-normal component points in the same direction as the surface-normal
vector, which allows the calculation of the vectorial surface-normal
component. Rearranging of vector multiplications yields the known form
gn=nb‖gn‖=nbnb⋅g=nbnb⋅g.
The surface-tangential component follows by subtracting the surface-normal
component from total gravitational acceleration:
gs=g-gn=g-nbnb⋅g=I-nbnb⋅g.
Movement in surface-normal direction is constrained by the basal topography,
which yields the basal pressure. Therefore, the surface-normal component
gn has to contribute to basal pressure pb
(Eq. ), and only the surface-tangential component
contributes to local acceleration ∂hu‾∂t (Eq. ). The total
gravitational acceleration can be reconstructed by summing up both
components:
g=gn+gs=nbnb⋅g+I-nbnb⋅g=I⋅g=g,
ensuring perfect conservation of three-dimensional momentum.
Splitting the divergence of a flux tensor
∇⋅M into a surface-tangential and surface-normal
part with simple projections to the surface-normal vector
nb.
The same concept can be applied to fluxes through the boundary of the control
volume, leading to the concept of surface partial differential operators
(∇s and ∇n).
Figure shows the divergence of a tensor,
∇⋅M, which could represent convective momentum
transport ∇⋅hu‾u‾ or the lateral
pressure gradient
12ρ∇pbh=12ρ∇⋅Ipbh.
Using Gauss' theorem, the divergence can be reformulated in terms of the
surface integral of face fluxes, which are defined
as the scalar product of the flux tensor M with the normal vector
on the face . In the discretised form,
integrals are replaced with sums over faces and in the case of SPDEs, volumes
collapse to surfaces, faces to edges and face fluxes to edge fluxes. For the
simple case, as shown in Fig. , we can write
∇⋅M=1Sbmout-min,
with area of the cell Sb and edge fluxes
min and mout. For the exact
formulation in terms of finite areas, the reader is refereed to
. Note that ∇⋅M is a
three-dimensional vector without any particular direction in relation to the
basal surface. Hence, it has a surface-tangential and a surface-normal
component which can be treated similarly to gravitational acceleration,
yielding the surface-normal component
∇n⋅M=nb‖∇n⋅M‖=nbnb⋅∇⋅M=nbnb⋅∇⋅M,
and the surface-tangential component
∇s⋅M=∇⋅M-∇n⋅M=∇⋅M-nbnb⋅∇⋅M=I-nbnb⋅∇⋅M.
Surface-normal and tangential components contribute to local acceleration and
basal pressure for reasons discussed in terms of gravitational acceleration.
Three-dimensional conservation is ensured for fluxes as well if
∇⋅M is calculated conservatively. Finally, we want
to note that velocity is a three-dimensional vector field and its direction
is not fixed a priori. However, velocity will always be aligned with the
surface because only surface-tangential components are present in the
respective conservation equation.
The supplement related to this article is available online at: https://doi.org/10.5194/gmd-11-2923-2018-supplement.
MR developed the model and the respective code. Simulations have been conducted by MR and AK.
AH covered GIS aspects and the production of respective figures. WF conceived the
presented investigation, verified the underlying analytical models and
supervised the project. All authors discussed the results and contributed to
the final paper.
The authors declare that they have no conflict of
interest.
Acknowledgements
We thank Mark Olesen and Andrew Heather (ESI-OpenCFD) for help regarding
OpenFOAM and a review of our solver code. We thank Matthias Granig and Felix
Oesterle (WLV) for support regarding SamosAT and for providing the respective
software. We thank our colleges, Iman Bathaeian, Jan-Thomas Fischer and
Fabian Schranz, for valuable comments on the manuscript. We thank the
OpenFOAM, ParaView and QGIS communities for sharing their code and providing
helpful advice. We further thank Stefan Hergarten, Julia Kowalski and one
anonymous reviewer for their valuable comments, which helped to increase the
clarity and quality of this paper. We gratefully acknowledge the financial
support of the OEAW project “Beyond dense flow avalanches” and the Vice Rectorate for Research at the
University of Innsbruck. The
computational results presented have been achieved (in part) using the HPC
infrastructure LEO of the University of Innsbruck. Edited by: Simone Marras Reviewed by: Julia
Kowalski, Stefan Hergarten, and one anonymous referee
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