The three-dimensional (3-D) modelling of water systems involving double-diffusive processes is challenging due to the large computation times required to solve the flow and transport of constituents. In 3-D systems that approach axisymmetry around a central location, computation times can be reduced by applying a 2-D axisymmetric model set-up. This article applies the Reynolds-averaged Navier–Stokes equations described in cylindrical coordinates and integrates them to guarantee mass and momentum conservation. The discretized equations are presented in a way that a Cartesian finite-volume model can be easily extended to the developed framework, which is demonstrated by the implementation into a non-hydrostatic free-surface flow model. This model employs temperature- and salinity-dependent densities, molecular diffusivities, and kinematic viscosity. One quantitative case study, based on an analytical solution derived for the radial expansion of a dense water layer, and two qualitative case studies demonstrate a good behaviour of the model for seepage inflows with contrasting salinities and temperatures. Four case studies with respect to double-diffusive processes in a stratified water body demonstrate that turbulent flows are not yet correctly modelled near the interfaces and that an advanced turbulence model is required.

Over the past decades, numerical salt and heat transport models
have increased their capability to capture patterns of double diffusion on
scales varying from laboratory set-ups to the ocean

Most numerical modelling studies of double-diffusive processes calculate
interfaces and salt and heat fluxes on the oceanic scale

Axisymmetric CFD models are applied in a wide variety of fields. Examples of
applications include the modelling of gas flow past a gravitating body in
astronomy

In some cases, axisymmetric grid set-ups can also be preferential for
hydrodynamic surface water models. Examples of such cases are
close-to-circular water bodies with uniform boundaries and the flow around a
central point (e.g., a local inflow from a pipe or groundwater seepage). The
occurrence of local saline seepage inflows into shallow water bodies of
contrasting temperatures has been described by

This article derives a framework for an axisymmetric free-surface RANS model,
which is implemented in SWASH. SWASH is an open-source non-hydrostatic
modelling code for the simulation of coastal flows including baroclinic
forcing

The development of an axisymmetric variation of SWASH falls in line with our research on localized saline water seepage in Dutch polders. To simulate the effect of a local seepage inflow on the temperature profile of the surface water body, a numerical model is required that accounts for sharp density gradients, a free surface, and potential double-diffusive processes. The axisymmetric grid set-up aids in correctly representing the volumetric inflow and modelling the flow processes around the local inflow. Rather than a fine-scale model that resolves all the relevant scales for double-diffusive processes, the current approach should be regarded as a pragmatic solution tool that can be run on relatively simple computer infrastructure.

In this article, we present the resulting numerical framework to extend a 2-D finite-volume model into a 2-D axisymmetric model by adding a few terms to the solution of the governing Navier–Stokes and transport equations. These terms are implemented in the SWASH code. The model code is further extended with a new transport module calculating salt and heat transfer. Although the model generally calculates with a mesh size that is larger than the size required to solve small-scale double-diffusive instabilities, the aim is to allow the model to approximate interface locations and salt and heat fluxes. The functioning of the code is validated with case studies involving different salinity and temperature gradients.

The governing equations in this study are the Reynolds-averaged
Navier–Stokes equations for the flow of an incompressible fluid derived in
cylindrical coordinates (

In these equations,

This RANS model allows for turbulence modelling with the standard

The pressure terms are split into hydrostatic and hydrodynamic terms
according to

Definition of the free-surface level

The free surface is calculated according to

The transport of mass and heat is calculated with the convection–diffusion
equation:

In the case that turbulence is modelled, the vertical turbulent diffusion,

In non-turbulent thermohaline systems, stability largely depends on density
gradients and molecular heat and salt diffusion rates, which in turn are
highly dependent on temperature and salinity. The heat and salt diffusivities
are related to temperature

At the free surface, we assume no wind and

Vertical grid definition (sigma layers)

A special case is the inner boundary at which symmetry occurs; for all
variables, the gradient is set to zero, except for horizontal momentum:

For the transport equation, a homogeneous Neumann boundary condition is
defined at each boundary (

The physical domain is discretized with a fixed cell width in the radial
direction. The width of the cells in the tangential direction increases by a
fixed angle

Axis definition.

For the vertical grid, sigma layering is employed, although part of the
layers can be defined by a fixed cell depth (Fig.

For reasons of momentum and mass conservation,

The momentum equations and the transport equation are integrated in a similar
fashion.

Since

We then spatially discretize the

Again, the alpha terms are marked with boxes. Another addition compared to
the Cartesian 2-DV solution are the

The governing equations are spatially discretized with a central differences
approach, except for the advective terms. The advective terms are discretized
with higher-order flux limiters

The horizontal time integration of the momentum and transport equations is
Euler explicit. The horizontal advective terms in the momentum equations are
solved with the predictor–corrector scheme of MacCormack

The numerical framework largely follows the SWASH solution procedure

This article validates the model qualitatively and quantitatively. The
behaviour of a local seepage inflow setting on double-diffusive layering is
verified qualitatively (Sect.

documented properties of systems of double-diffusive convection
and salt fingering (Sect.

and the expected expansion of an unconditionally stable layer near
a bottom seepage inflow, for which an analytical solution is derived in
Sect.

In all the case studies (Table

The dimensions, properties, and consequent stability parameters
applied in the case studies.

Defined cell depths for Cases 1 to 7. For plotting reasons, the
vertical axis displays the depth from the water surface relative to the local
water depth. The cell depths that are defined relative to the local water
depth (as marked by

This subsection lists several common metrics, which we applied to
quantitatively validate our simulations of double-diffusive systems with
varying density gradients (Cases 1 to 4). To validate the applicability of
the standard

The stability of a double-diffusive system is commonly expressed by its
Turner angle

The expansion coefficients

Cases 1 and 2 concern a system with two layers of equal depth in which a
cold and fresh water layer is overlying a warm and saline water layer
(Table

Based on the Turner angles of Cases 3 and 4 in which warm and saline water
is overlying cold and fresh water, salt fingers are expected to occur
(Table

An effective transport of heat and salt over the interface while maintaining
a sharp interface is expected as this is a known property of double-diffusive
salt fingers

A clear definition of the interface location is relevant for the
determination of the boundary properties and the heat and salt flux across
the boundary. In each simulation, the interface location

Interface positioning over time displayed on the depth profiles of

The vertical saline and thermal density fluxes across the interface,

The simulated salt and heat fluxes are compared with theoretical fluxes based
on molecular diffusivities and double-diffusion-specific eddy diffusivities
according to the equation

For double-diffusive convection, we also compare the heat fluxes with
theoretical heat fluxes as predicted by

For systems of double-diffusive convection, we calculate the evolution of the
boundary layer thicknesses

As a last validation metric for the salt-finger cases, we employ the Stern
number

Stern suggested that the growth of salt fingers is arrested when

The quantitative validation of an unconditionally stable bottom layer is
based on an analytical solution for the radial expansion of this dense layer
from a central inflow under laminar flow conditions (Case 5;
Table

In this quantitative case study, the central inflow has an outer radius of
0.2 m. To allow for a slow development of the bottom layer, the inflow is
placed slightly deeper compared to the rest of the bottom, and the inflow
velocity linearly increases over the first 20 min. The discharge over the
right outflow boundary is set equal to the inflow discharge:

Conceptualization of the quantitative validation (Case 5), with locations of the salinity and temperature interfaces at a certain time after the start of a central inflow. The inflow is colder and more saline than the overlying water body.

To derive the growth rates of the temperature and salinity interfaces, we
consider the similarity solution of the heat equation for a fixed boundary
concentration

Derivation over time results in the time-dependent mass flux over the
interface:

We assume that no mass is stored in the lower layer. Consequently, the mass
flux that crosses the interface is equal to the net mass flux into the domain

This equation can be used to validate the interface growth of both the salinity and temperature interface in the case of laminar flow.

Double-diffusive convection in a layered system with a cold and
fresh layer on top of warm and saline water (

Time evolution of the interface thicknesses

Time evolution of the salt (red) and temperature fluxes (blue) over
the interface for

Cases 6 and 7 represent seepage inflows similar to the ones for which
this modelling approach is developed. A dual-layered system is built up by a
central inflow through the bottom with an outer radius of 0.25 m
(Table

The performance of the numerical framework was tested in several case studies
subject to double-diffusive processes. The numerical results of these case
studies and the extended SWASH code are presented in

The temperature and salinity gradients in Cases 1 and 2 yield a
theoretical onset of double-diffusive convection, with respective Turner
angles of

The boundary layer thickness ratio

Simulated and theoretical heat fluxes according to
Eqs. (

Regarding the tendency to reach a steady state step by step by building up a
system with a stable boundary layer, our findings for Case 1 with a
turbulence model seem more alarming. Here,

The simulations for Case 2 also show an initial increase in

The poor performance of the standard

The ratio of the simulated heat fluxes to the

In line with the expectations for turbulent flows, the simulations for Case 2
show a large variation in heat and salt transport (Fig.

The dissimilar behaviour of our simulation with a turbulence model can be
explained by the employed eddy diffusivities, which have similar values for
salt and heat diffusion (note that the turbulent Prandtl and Schmidt number
have similar values). These eddy diffusivities were not employed in the
simulation without a turbulence model, which indicates that the similar heat
and salt transport across the interface is caused by turbulent mixing through
this interface. We refer to Sect.

Salt fingering in a layered system with warm and saline water on
top of a cold and fresh layer (

The numerical results for Case 3 (

Based on the difference in density ratios, the salt fingers in Case 4 are
expected to transport more salt and heat than Case 3 (Sect.

Similar to Cases 1 and 2 (Fig.

The evolution of flux ratios over time for

Based on a 3-D DNS model,

The evolution of the Stern numbers over time for

Evolution of the interface between a warm and fresh water body and a
bottom cold and saline layer developing form a central inflow (Case 5). After

The analytical solution for the radial expansion of inflowing cold and saline
water (Eq.

Accounting for a purely molecular diffusion, the numerical results show a
fair agreement with the analytical results. As we found some small occasional
eddies occurring after

One critical note here is the sensitivity of the interface growth to the
definition of the interface location. Similar to the previous cases, we
defined the interface location as halfway to the step change between the inflow
concentration (

The temperature and salinity gradients in Case 6 yield the onset of
double-diffusive convection. Like Cases 1 to 4, a sharp interface
develops over which salt and heat is transported by diffusion.
Figure

Double-diffusive layering (Case 6) with cold and fresh water on top
of a warm and saline inflow (

Unstable system (Case 7) with denser cold and fresh water on top of
a warm and saline inflow (

Compared to Case 6, a slightly altered inflow temperature and salinity in
Case 7 theoretically makes the developing layer gravitationally unstable
(Table

Interestingly, plumes develop from the upward flow. Downward plumes are also visible below the floating warm and saline water. Like the salt fingers in Cases 3 and 4, in which warm and saline water also overlaid cold and fresh water, this is a mechanism to dissipate the heat and salt gradients.

In the previous subsections, we found that the standard

In Sect.

More advanced turbulence models have been developed for systems with large
density gradients

This article reports the successful derivation of an axisymmetric framework for a hydrodynamic model incorporating salt and heat transport. This model set-up allows us to efficiently calculate salt and heat transport whenever a situation is modelled that can be approximated by axisymmetry around a central location. The 2-D axisymmetric grid description demands approximately the same execution time as a regular 2-DV description with the same dense mesh and therefore avoids the need to solve the equations over a dense mesh in the third spatial dimension.

For our purpose of studying shallow water bodies, three aspects were important: (1) the inclusion of a free surface, (2) the efficient solution of a circular seepage inflow, which makes the problem three-dimensional, and (3) a proper simulation of density-driven flow and double-diffusivity-driven salt and heat transport. The former aspect was already fulfilled by employing the SWASH framework.

The second aspect was solved by assuming axisymmetry for the Reynolds-averaged Navier–Stokes equation in cylindrical coordinates. The derived numerical framework is presented as a Cartesian 2-DV description with a few additional terms and width compensation factors. Our implementation of these terms in the non-hydrostatic SWASH model demonstrates the opportunity to easily extend a 2-DV model towards the presented 2-D axisymmetric model.

The third aspect was fulfilled by extending SWASH with a new density and
diffusivity module. The case studies demonstrate explainable behaviour for
density-driven flow and double-diffusivity-driven salt and heat transport.
The formation of convective layers and salt fingers is in
accordance with the theory of double diffusivity and the enhanced
salt and heat fluxes across the interface for density gradients approaching
unity. Other validation metrics show that the RANS model does not meet the
expected flux ratios and stability criteria in all cases, which is
hypothesized to be caused by a defective turbulence modelling for systems of
large density gradients. Replacing the standard

An analytic validation method was presented to evaluate the model's performance for a cold and saline inflow developing a dense water layer near the bottom. For laminar flow conditions, the numerical model showed a similar radial expansion of the bottom layer as expected from analytical results. Although the model is already able to show expected behaviour in the double-diffusive regime, we recommend a further exploration of its limitations and possibilities. For example, a grid convergence study should indicate whether the selected mesh size yields a convergence of results for all diffusion- and advection-dominated cases. Further, a nearer comparison with DNS model results would support the validation of the model. In future applications, we stress that this model approach should be employed as a RANS model that simulates thermohaline stratification processes on a larger scale. As such, the model can be favourable in applications that allow for an axisymmetric approach.

The model data for the five case studies and the
extended SWASH code are accessible at

When the continuity, momentum, and transport equations are integrated over the
cell depth, the Leibniz integral rule is applied to the time derivatives and
the horizontal spatial derivatives. Here, we show the cell depth integration
of

The derivatives

The authors declare that they have no conflict of interest.

This project has been funded by the Netherlands Organisation for Scientific Research (NWO), project number 842.00.004. Edited by: Ignacio Pisso Reviewed by: three anonymous referees