4.1. Viscoplastic Flow around a Sphere
We first validate the ability of the mudMixtureFoam solver to simulate the rheological properties of a Bingham fluid passing over a stationary spherical particle. This is carried out by comparing the result of mudMixtureFoam to the work of Gavrilov et al. [
24], who modeled a Bingham plastic flowing past a sphere with a different finite-volume solver. The properties that we have chosen to compare are the surface drag over the sphere and the viscous stress distribution in the Bingham fluid in the vicinity of the sphere surface, two properties of the fluid that depend heavily on the rheology.
Following the methodology of Gavrilov et al. [
24], we utilize radial symmetry to minimize the computational domain necessary to model. We construct a 3D domain of extent
with a solid spherical surface centered at
. This results in a quarter of the solid sphere extending into our domain (
Figure 1). Using radial symmetry, we can extrapolate that the unmodeled volumes are symmetrical reflections across the y = 0 plane and the z = 0 plane centered at
.
The domain size is set to cover a distance 50 radii from the sphere center. A circumferential resolution of 72 cells is defined at the surface of the sphere (
Figure 2), with this high-resolution area extending to a distance five radii from the sphere center. A constant fluid flow aligned along the x-axis is used to initialize the simulation and is also imposed as an inlet boundary condition with magnitudes indicated in
Table 1. The surface of the sphere is set to be a no-slip wall boundary, with the other domain boundaries being assigned the symmetry boundary condition. The forward Euler scheme was used for time integration of the governing equations. The simulated flow field quickly converged to the final steady state for the low Reynolds number considered in the present tests.
Drag and viscosity are intimately related to the system rheology. As fluid flow encounters the solid sphere and is deflected around the sphere surface, the no-slip wall boundary creates a shear stress around the sphere, which decreases the viscosity of the Bingham fluid in the region where the yield stress is exceeded. Using his simulations Gavrilov [
24] obtained the following empirical drag force on the sphere
as a function of the Reynolds number (
) of the fluid
and the Bingham number (
) of the fluid
where
D is the sphere diameter, V is the velocity of the fluid,
is the fluid density,
is the dynamic viscosity, and
is the yield stress; the last three properties would be functions of the sediment concentration in the case of mud suspensions.
We considered three unique cases of steady Bingham fluid flow over a range of
and Reynolds numbers from
(shown in
Table 1). Sphere diameter and fluid velocity were varied to obtain a range of Reynolds and Bingham numbers, while the yield stress, density, and viscosity were held constant.
Drag-induced shear causes stress near the sphere surface, which in turn results in a lowered effective viscosity of the Bingham fluid near the sphere surface. As the Bingham number decreases, or Reynolds number increases, we expect that the flow and shear effects would more strongly dominate, resulting in a larger area of lowered viscosity around the sphere (as observed in
Figure 3,
Figure 4 and
Figure 5). These figures represent contour cross-sections of the fluid viscosity through the center of the rigid sphere.
Figure 3 and
Figure 5 correspond directly to cases A and C, respectively, as examined by Gavrilov et al. [
24]. Case B (
Figure 4) was undertaken to demonstrate the ability of mudMixtureFoam to handle fluids with very large Bingham numbers.
The contour lines correspond to the following range of normalized values of effective viscosity: . In case A, the region of lowered viscosity forms a symmetrical bubble around the rigid sphere, with a smaller ring of higher viscosity halfway around the sphere. As the Bingham number increases, the region of lowered viscosity deforms somewhat, extending outwards in the upstream and downstream directions, and a distinct divot forms at the top. Presumably, this can be attributed to the effects of increased yield stress causing the fluid to resist shear near the sphere surface. Predictably, this causes the region of lowered viscosity to have a more limited extension in the cross-stream direction away from the sphere. When the Reynolds number is dramatically increased, as in case C, the region of lowered viscosity can be seen to lengthen quite dramatically downstream. This is expected, as at high Reynolds numbers, the inertial forces are strong enough to overcome the viscous forces, and the fluid stress exceeds the yield stress of the Bingham fluid, both near the sphere surface and downstream of the rigid sphere, creating a wake of lowered viscosity.
Figure 6 and
Figure 7 display the vertical and horizontal cross-sections of viscosity taken from cases A and C, compared with the contour profiles of data digitized from the corresponding Gavrilov results.
The profiles of viscosity through the sphere determined by the mudMixtureFoam solver show a good match with the simulation results from Gavrilov et al. [
24]. A quantitative comparison of the viscosity profiles to Gavrilov et al. [
24] was made by calculation of the root-mean-square error (RMSE), normalized using the viscosity range. Case A shows the best fit of the model with the established simulation, with a normalized RMSE of 0.0135 in the vertical and 0.0011 in the horizontal. Case C shows an excellent match in the vertical, with a normalized RMSE of 0.000237. The horizontal profile of Case C shows a good match on the upstream side (normalized RMSE = 0.000379), but on the downstream there is a clear under-prediction of the extent of the region of lowered viscosity. This resulted in a normalized RSME value of 0.195. This is a notable divergence from the Gavrilov simulation results, but was likely a result of the limited extent of downstream refinements used within the OpenFOAM domain.
Successful modeling of drag force by mudMixtureFoam would produce values for drag force over the sphere surface consistent with the theoretical drag force obtained from Equation (
11).
Figure 8 demonstrates that the modeled value of the drag converged toward the empirical value over the last several hundred model iterations of the test case runs.
Taken together, the examination of drag and viscosity clearly demonstrate that the new mudMixtureFoam solver is able to reliably model the rheology of a Bingham plastic fluid, making it a suitable method for modeling areas with dense muddy sediment. With this validation of the rheological component of the model, we turn to the examination of the ability of mudMixtureFoam to appropriately handle the generation of turbulence in the bottom boundary layer.
4.2. Turbulence in Single-Phase Oscillatory Boundary Layer
The ultimate goal of the mudMixtureFoam solver is to model turbulent resuspension of sediments and their transport in the bottom boundary layer. There is a large body of simulation work concerning bottom boundary later turbulence under oscillating flow conditions [
15,
19,
20,
34,
35], and we have selected here the work of Costamagna et al. [
36] to serve as a benchmark for our model. There are two facets to the motivation for this section. The first is to show that we can replicate the growth and behavior of intermittent turbulence in the bottom boundary layer, and the second is to test the stability of our solver for non-solenoidal flow under conditions of non-uniform sediment concentration.
We consider a 3D domain (a small box near the bottom boundary) analogous to that used by Costamagna. The domain extent is a multiple of the Stokes boundary length scale
where
is the angular frequency of the oscillatory flow. Costamagna et al. [
36] tested the domain size necessary to produce turbulent effects, and a domain with dimensions
was sufficient to produce a good match when compared to experimental observation [
37]. While subsequent studies have expanded this domain to produce more targeted turbulence effects, such as tracking turbulent spots within the flow [
34], for a simple comparison we deemed the smaller domain sufficient. Periodic boundary conditions were selected on the horizontal boundaries, a no-slip wall condition was forced at the bottom boundary, and a slip boundary condition was selected for the upper boundary.
Flow through the domain was forced by the oscillating pressure gradient term
where
is the angular frequency of the oscillating wave, given by
and where T is the period of the wave. This term corresponds to a free stream wave with velocity
. A period of five seconds was selected, resulting in an angular frequency
s
and Stokes boundary layer thickness
mm. In oscillatory flow, the flow regime of laminar, disturbed laminar, intermittent turbulent, or fully turbulent is typically described using the Stokes Reynolds number [
38,
39]
where the intermittently turbulent regime exists in the range of
. To keep consistency with with energetic shelf wave conditions, and to produce Stokes–Reynolds numbers of the same level as those used by Costamagna et al. [
36], a
of
m/s was selected. The natural transition to turbulence from laminar flow can be triggered by initially small variations induced by perturbations, such as wall waviness or vibration. These cause small variations from the laminar regime, which grow and cause the breakdown of laminar flow into turbulence. An infinitesimal waviness comprised of a superposition of sinusoidal waves was applied to the bottom boundary of our domain
where
and
represent the nondimensionalized
x and
y positions, respectively, and
is the vertical position of the bottom boundary. This expression is characterized by the wavenumbers
in the x dimension and
in the y dimension. The parameters for waviness used in this study are given in
Table 2.
Taking into account all the factors above, we chose our domain to be of size 6.33 cm × 3.17 cm × 3.17 cm, with
grid points. The infinitesimal bottom boundary waviness described by Equation (
17) was introduced by calculating a spline fit through a series of points.
We studied two specific case runs. The first had a zero sediment concentration throughout the domain, for a direct comparison of our model results to those of Costamagna et al. [
36]. The second case introduced a very small sediment concentration, initialized using the exponential profile:
to demonstrate that the solver can perform calculations with concentration-dependent terms. In the intermittently turbulent regime, the turbulent kinetic energy (TKE) is characterized by an oscillation, where turbulence grows during the early part of the deceleration phase and diminishes during the early phases of accelerating flow.
Figure 9 shows the free-stream mean velocity, the oscillating pressure gradient forcing term, and the domain-averaged turbulent kinetic energy. The first simulated cycle served as a ramp-up of the model, while phase averages were taken over the second and third cycles. We begin with a quiescent fluid with an initially large pressure gradient, which follows a sinusoidal behavior through one cycle. This results in a velocity oscillation which has the expected
phase shift. For our five-second cycle, this means we expect maximum velocity magnitudes at
and
of each cycle, which can clearly be seen in
Figure 9. Turbulent kinetic energy follows the velocity oscillations, reaching maximum values at the early phases of decelerating cross-flow.
Figure 10 shows a comparison between both mudMixtureFoam solver case runs and the digitized results of Costamagna et al. [
36]. The streamwise velocity fluctuations normalized by
decreased to values lower than 0.01 above 20 mm from the bottom. At
, where the oscillating flow is zero and the acceleration is negative, the maximum fluctuations occur between 5 mm and 10 mm, reaching values of up to 0.05 with regularity. At 1.24
, when the oscillating flow is oriented in the negative x direction, peak fluctuations occur around 1 mm from the bottom (comparable to
), reaching values of just under 0.09. At 1.64
under conditions of negative flow and positive acceleration, the profile of streamwise fluctuations is seen to increase in the region below 15 mm, reaching values up to 0.12 near the bottom. Normalized vertical velocity fluctuations have similar behaviors, where at
, the profile can be seen to have a large increase in values under 25 mm, with the largest values occurring between 15 mm and 5 mm, reaching magnitudes of just under 0.04. At 1.64
, the fluctuations can be seen to increase towards the bottom, reaching a peak at 3 mm (about
) over 0.04.
Figure 10 also shows that adding a small concentration profile did not have a significant impact on the momentum of the solution, which can be seen in how closely it follows the case without sediment. The results also compare favorably with the experimental work of Jensen et. al. [
37] on oscillatory flows in an intermittently turbulent regime (See
Figure 10, diamond symbols). Doubling the grid resolution in the simulation only had a small effect, such that the root-mean-squared velocity perturbations in the streamwise direction did not exceed
at the phase of maximum flow (see for example,
Figure 10, first panel). This validates the ability of our solver to properly handle sediment concentrations within the bottom boundary layer under oscillating flow.
The volume fraction field is a tracer and its behavior through the domain provides insight into the development of turbulence over time.
Figure 11 shows the volume fraction on an x–z plane cross-section through the domain. From this, the time evolution of turbulence can be seen, as the sediment goes from a well ordered initial gradient (
Figure 11a,b) towards well-mixed homogenized conditions (
Figure 11d).
Figure 12 shows the growth and oscillations of the domain-averaged turbulent kinetic energy generated with the mudMixtureFoam solver through the first three flow cycles. Growth of turbulent kinetic energy occurs during the first cycle and quickly stabilizes for the second and third cycles. As expected, the small amount of density gradients in the test with the concentration profile did not alter significantly the final equilibrium value of the turbulent kinetic energy. These results demonstrate that the mudMixtureFoam solver can reproduce the generation of turbulence within the bottom boundary layer, as well as handle small concentrations of sediment, without having a substantial effect on the solution of the momentum.
4.3. Two-Phase Flow in Oscillatory Boundary Layer
Here, we replicate the simulation of suspended fine sediments under an oscillatory flow from Ozdemir et al. [
19]. In their methodology, the Boussinesq approximation was assumed for fluid flow, and the resulting momentum equation was used to formulate a model designed to capture the interactions of sediment properties and turbulence in the bottom boundary layer. Reproducing these numerical cases will give further validation to the applicability of our non-Boussinesq model to fluid flows under such conditions, and facilitate the expansion of the modeled conditions to include denser suspensions.
An oscillating free-stream velocity of 0.56 m/s and a period of 10 s were selected to simulate highly energetic shelf conditions. This resulted in a Stokes boundary length of 1.8 mm and a Stokes–Reynolds number of 1000. The domain used in the previous section was extended vertically to achieve consistency with Ozdemir et al. This extension resulted in a domain with a vertical span of 60 Stokes boundary lengths. Length scales were normalized using this boundary layer thickness, and velocity scales were normalized using the maximum free stream velocity,
. Ozdemir et al. [
19] made use of a pseudo-spectral method to resolve the initialization of turbulence at the bottom boundary, which we cannot reproduce with our second order spatial accuracy. Instead, we made use of the bottom waviness (Equation (
17)) approach that was utilized by Costamagna et al. [
36] and Mazzuoli et al. [
40] to provide the infinitesimal “kick” that triggers the natural transition to turbulence.
The domain averaged sediment concentration was held to be 0.001 by volume within the domain, and initialized with the exponential profile , where c is the volumetric concentration of sediment, and z is the non-dimensional vertical position vector. A bulk Richardson number of , and Stokes–Reynolds number of 1000 were chosen for consistency with Ozdemir’s numerical simulations.
We ran cases 2 and 3 from
Table 3 with the mudMixtureFoam solver and compared the results to the digitized data output of Ozdemir et al. [
19]. Points of comparison are the behavior of the along-stream velocity profiles and the profiles of concentration. Values are taken on three different phases of the oscillating cycle:
,
, and
. These phases correspond to the decelerating phase, the zero freestream velocity phase, and the negatively accelerating phase, respectively.
The phase-averaged concentration profiles in our simulations (blue lines) are in good agreement with those of Ozdemir et al. (red dots) and formed a lutocline, a sharp concentration gradient, at approximately 25 Stokes boundary lengths from the bottom (
Figure 13 top row). The lutocline separates the upper fluid layer (with low magnitudes of turbulence) from the lower turbulent layer. Likewise, our simulations reproduced the vertical behavior of turbulence in relation to the lutocline, where the region below the lutocline is characterized by substantially increased values of turbulent velocity fluctuations (
Figure 13 bottom row). The presence of a distinct lutocline is a characteristic behavior of the intermittently turbulent flow regime. The flow in this layer is highly energetic, displaying high values of streamwise root mean squared velocity, as can be seen in the bottom row of
Figure 13. We found a similar agreement between our model and Ozdemir et al. for case 3 in
Table 3.