The effect of Schmidt number on gravity current flows: The formation of large-scale three-dimensional structures

Gravity currents are primarily horizontal flows arising from a density difference between the current and surrounding ambient fluids. This density difference occurs with variation in concentration of solutes or suspended sediments or temperature difference between the current and the ambient fluids.^1,2 Gravity currents are a common class of flow, with examples including thunderstorm outflows and powder snow avalanches,^1,3 and are one of the primary mechanisms of sediment transport in oceans.^4,5 There exists extensive research into the dynamics of gravity currents, including numerical investigations.^6–13 The typical structure of gravity current flows as described by Kneller and Buckee¹⁴ is presented in Fig. 1 and consists of a head region followed by a body. This body can be divided into a dense lower layer overlaid by a less-dense layer of mixed current and ambient fluids. Despite the body typically forming by far the largest part of gravity current flows,^15–17 the existing research has primarily focused on the head of the flow. Additionally, numerical work has typically assumed a Schmidt number of 1 to investigate flows with $Sc=O(1000)$⁠.^{6,9,11–13,18–24} Therefore, characteristics such as the effect of Schmidt number on the flow, and the nature of large-scale structures within the body, remain poorly understood.

Gravity current properties are characterized using a small number of dimensionless parameters:¹⁴ the Reynolds, densimetric Froude (affected by the bed slope²⁶), gradient Richardson, and Schmidt numbers. The Reynolds number is defined as the ratio of inertia to viscous forces (⁠$Re=UcLcν$⁠, where U_c and L_c are velocity and length scales that are characteristic of the flow, and ν is the kinematic viscosity of the fluid), and increasing Reynolds number reduces the size of the smallest length scales within the flow that must be resolved while conducting direct numerical simulations (DNS).²⁷ Decreasing Reynolds number, below the point of similarity (⁠$Re≈1000$⁠),¹ also changes the nature of the current head including the formation of Kelvin–Helmholtz billows and the over-riding of ambient fluid leading to lobe-and-cleft structures.^1,28,29 Re significantly affects rates of mixing and entrainment, with increasing Re dictating whether the primary mixing mechanism is Holmboe waves, Kelvin–Helmholtz vortex rolls, or Kelvin–Helmholtz billows.^30,31 The Schmidt number, analogous to the Prandtl number, is defined as the ratio of momentum and mass diffusivities (⁠$Sc=ν/D$⁠, where D is mass diffusivity) and is a key parameter in understanding mixing on the molecular level.^32,33 As Schmidt number increases, diffusion decreases and momentum becomes the dominant mass transfer mechanism.

Increasing the Schmidt number from 1 reduces the length scales associated with mixing within the flow, reducing the smallest lengths that must be captured from the Kolmogorov scale (η_K) to the Batchelor scale (⁠$ηB=ηKSc−1/2$⁠),^34,35 and mixing is expected to decrease.^32,36 It has been demonstrated that the finer-scale structures that develop with increasing Schmidt number may cause larger scale structural changes in the flow, such as increased plume length in turbulent jets,³² and stronger three-dimensional motions resulting from changes to density profiles and stronger density gradients.^36–38 

The value of Schmidt number varies dramatically depending on the source of the density difference, for example being $O(1)$ for temperature-driven flows, $O(1000)$ for solutes in water, and being strongly dependent on grain size but often much larger for sediment in water [for example, being $O(109)$ for 100 $μ$m sand].^{8,33,34,39–41} Despite this, existing numerical investigations into gravity current structure typically assume Sc = 1.^{6–10,33,42,43} The large computational cost involved in resolving the Batchelor scale (the cost of direct numerical simulation scaling with $Re3Sc2$⁠)^27,34 means that few works have so far considered the effect of Schmidt number on gravity current flows. Birman et al.⁴² and Necker et al.⁴³ justify their use of Sc = 1 through test calculations that suggest Schmidt number has little influence on gravity current structure for $Sc=0.2→5$⁠, while Härtel et al.⁴⁴ claim that Schmidt number dependence is weak unless Sc is very small (⁠$≪1$⁠). Ooi et al.⁴⁵ use comparison of two two-dimensional large-eddy simulations of lock-exchange type flows, with a similar comparison of two three-dimensional large-eddy simulations in Ooi et al.,⁴⁶ to conclude that Schmidt number has only a small effect on properties such as current front velocity and shape. The effect of Schmidt number on other flow features is not considered, nor the combined effects of Reynolds and Schmidt number. Deepwell and Stastna,⁴⁷ in their study of flows consisting of a gravity current traveling along a pycnocline(s), conclude that the mass transport capability of internal solitary-like waves increases as Schmidt number is increased between 1 and 20, but does not continue to change when Schmidt number is further increased to 40 (suggesting that exact Schmidt number matching may not be required to capture flow dynamics). To date, the only research including a parametric study investigating the effect of Schmidt number on gravity current flows is the work of Bonometti and Balachandar.³³ 

Bonometti and Balachandar³³ use a combination of a pseudospectral method and a finite-volume/volume of fluid interface capturing method to investigate the parameter space $1≤Sc≤∞$ and $100≤Re≤10 000$⁠. For Re = 10 000, they conclude that neither the front velocity nor the level of mixing is strongly dependent on Sc, though decreasing Sc does increase the thickness of the layer of mixed ambient and current fluids. For the lower Reynolds number flows, they observe that increasing Sc changes head shape, with a depression separating head from body appearing as Sc increases, and that the effect of Sc on front velocity in these flows is highly dependent on the density contour chosen to define the front. They also claim that while the pattern of lobe-and-cleft structures is not strongly dependent on the Schmidt number, the formation of vortices along the body is. A scatterplot based on a table from Bonometti and Balachandar³³ is presented in Fig. 2, showing the distribution of stable/unstable interfaces between the current and ambient fluids based on a bulk Richardson number for their work and a few other investigations. This suggests that the interface stability is only weakly dependent on Schmidt number, with the interface becoming slightly more stable with increased Sc. However, these data are based almost entirely on two-dimensional direct numerical simulation (DNS) datasets. The flow structure resulting from two- and three-dimensional simulation of gravity currents with Reynolds numbers 317 and 10⁴ from the work of Bonometti and Balachandar³³ is presented in Fig. 3. While the two-dimensional work captures some features well, others require consideration of three-dimensional flow. Two-dimensional DNS of gravity currents is not able to reproduce large-scale coherent motions and three-dimensional flow features, such as the formation of lobe-and-cleft structures and the breakdown of interfacial billows, capture the spanwise dissipation of energy, or accurately estimate the energy budget of the flow.^{8,9,20,22,48–50} Therefore, three-dimensional simulations are needed to establish the effect of Schmidt number on three-dimensional flow features and to confirm the relationship between interface stability and Schmidt number.

In this work, three-dimensional direct numerical simulation (DNS) is used to investigate the structure and dynamics of constant-influx solute-based gravity current flows by providing the instantaneous density and velocity fields. As well as considering the effect of Reynolds number, the impact of varying the Schmidt number on both the head and body of gravity current flows will be investigated. The work addresses some of the remaining questions regarding how reasonable an assumption of Sc = 1 is for such flows, specifically the key aims are to discuss: (i) how Reynolds and Schmidt numbers affect the structure of the head, in particular, the formation of lobe-and-cleft structures, (ii) how Reynolds and Schmidt numbers affect the structure of flow behind the head, in particular, the stability of the current–ambient interface, (iii) which of the changes observed with increased Schmidt number also occur with increased Reynolds number, and (iv) when assuming a low Schmidt number to reduce the computational cost of numerical investigations may be justified.

The spectral element solver Nek5000⁵¹ is used to simulate three-dimensional gravity current in the domain illustrated in Fig. 4. This domain is designed to closely reproduce the experimental domain presented in Marshall et al.,²⁵ Marshall,⁵² with a simplified outlet. The governing equations are the nondimensional, incompressible, Boussinesq Navier–Stokes, salinity, and continuity equations,

where U is the velocity vector, t time, $P=p+ρagY$ where P is the kinematic pressure field, τ the stress tensor, $Re=UcLc/ν$ the Reynolds number, ν the kinematic viscosity, $Frd=Uc/g′Lc$ the densimetric Froude number, $Sc=ν/D$ the Schmidt number, D the mass diffusivity, g and $ĝ=(sin θ, cos θ,0)$ the magnitude and direction of the gravitational acceleration, θ bed slope, S salinity [with $ΔS=(S−Sa)$], $(·)a$ indicates a property of the ambient fluid, and $(·)̃$ indicates a dimensionless variable. As described in McWilliams,⁵³ here a simplified linear dependence of density on salinity has been employed,

where ρ is density, and $β=1ρ∂ρ∂S$ the haline contraction coefficient. The dimensionless variables are defined relative to the characteristic length, velocity, and time scales as shown in Table I. Time advancement is performed using a semi-implicit scheme that combines the implicit third-order backward difference and the explicit third-order extrapolation schemes, as described in Mittal et al.⁵⁴ Spatial discretization is based on the high-order spectral element method.^55–57 To ensure sufficient resolution, the wall $y+, x+,$ and z⁺ values (defined as $y+=uwy/ν$ where y is the distance to the nearest wall, and u_w the wall friction velocity with equivalent statements for the other spatial dimensions) are kept below 0.05 for the nearest grid point to the wall, and the first ten points are within $y+<10$⁠.^58,59 Additionally, decay of several orders of magnitude is observed in the energy spectrum for all variables.^6,60

The characteristic length L_c, velocity U_c, and time t_c scales are defined a priori. The characteristic length scale for all cases is chosen to be the height of the internal fitting to limit initial flow height, $Lc=0.05 m$ (refer to Fig. 4). For the highest Reynolds number case, the viscosity is chosen to match that of the experimental work in Marshall et al.,²⁵ Marshall,⁵² $ν=1.09×10−6 m2 s−1$⁠, and the characteristic velocity, $Uc=0.065 ms−1$⁠, is chosen to give the desired input Reynolds number Re_I = 3000. This represents an average velocity under the initial internal fitting. The characteristic timescale $Tc=Lc/Uc=0.77 s$⁠. To reduce the Reynolds number, the characteristic velocity scale is kept constant and the fluid viscosity varied. Schmidt number is varied by changing mass diffusivity, D. The input densimetric Froude number, $FrD,I=Uc/g′Lc=0.54$⁠, is the same for all cases, as is the source Froude number calculated using the inlet velocity (⁠$UI=0.22 ms−1$⁠) and inlet diameter (⁠$LI=0.0254 m$⁠) as characteristic velocity and length scales, $FrD,S=UI/g′LI=2.54$⁠. The input parameters for each case are shown in Table II.

The inlet flow is matched to Marshall et al.²⁵ and Marshall.⁵² The maximum inlet velocity in this work, U_I, is approximated by dividing the lowest influx value from Marshall et al.²⁵ (⁠$7×10−5 m3 s−1$⁠) by the inlet area (⁠$3.34×10−4 m2$⁠). The inlet has a dimensionless radius of 0.254 and is covered with a coarse mesh with holes of dimensionless radius 0.078. These holes are centered at locations,

with the inlet velocity approximated by

where r varies from 0 to 1 from the center to the edge of each small circle. Where the velocity on the inlet is nonzero the salinity $S̃=1.03$⁠, compared to $S̃=1.00$ in the rest of the domain, and therefore $ΔSI=0.03$ with a haline contraction coefficient of β = 1 to achieve a density difference of 3% between the current and ambient fluids. The outlet is approximated by a square outlet placed in the bottom right corner of the domain, with a special boundary condition implementation that matches the total outflow to the net inflow into the domain. In this work, X, Y, and Z correspond to downstream, vertical, and cross-stream directions with corresponding downstream, vertical, and cross-stream velocities U, V, and W.

The effect of Schmidt number on the density structure of the flow can be established by considering contours of proportional excess density (⁠$ΔS̃$⁠). Figure 5 shows contours of $ΔS̃$ at $t̃=23.4$ at a central cross-stream location. The rightmost column of this figure shows two contours for each case—that with density just above the ambient density, $ΔS̃=0.03$⁠, and that with density halfway between the densities of the ambient fluid and that pumped in at the inlet, $ΔS̃=0.48$⁠. The contour with $ΔS̃=0.03$ is considered to be the current boundary. Increasing Schmidt number has little impact on current front velocity over the short time frame considered here, though increasing Reynolds number increases front velocity as expected. Increasing either Reynolds number or Schmidt number leads to a depression behind the head, angled toward the current front.

For each case, the current height (defined as the height of the $ΔS̃=0.03$ contour at the left-most point in Fig. 5), the thickness of the mixed region (defined as the difference in height of the $ΔS̃=0.03$ and $ΔS̃=0.48$ contours at the left-most point of the figure), and the difference in front positions of the $ΔS̃=0.03$ and $ΔS̃=0.48$ contours are estimated by inspecting Fig. 5, and listed in Table III. The height of the head is approximately constant across cases with constant Schmidt number. For Sc = 1 the head height is $Ỹ≈1.1$⁠, reducing to $Ỹ≈0.8$ for Sc = 10. Increasing Schmidt number consistently reduces body height, the thickness of the mixed layer as a proportion of total flow height (Fig. 6), and the difference in the front position of the two contours (illustrating that increasing Schmidt number leads to dense fluid reaching closer to the front of the flow).

The height of the $ΔS̃=0.48$ contour, however, is not strongly affected by increasing Schmidt number, suggesting that evidence of the effect of Schmidt number on quantities such as current height or front velocity is highly dependent on the contour chosen to define the current boundary. In fact, the height of this contour is the same across all cases except Re_I = 3000, suggesting a greater degree of mixing in this case. This is likely an artifact of relatively short simulation duration. Longer simulation time will result in changes in contours further from the interface. Increasing Schmidt number reduces current height, with a 33% decrease between the $(ReI,Sc)=(100,1)$ and (100, 10) cases [with a further decrease of 25% between $(ReI,Sc)=(100,10)$ and (100, 100)], though this decrease is reduced to 25% between $(ReI,Sc)=(500,1)$ and (500, 10). Similarly increasing Schmidt number reduces the percentage of the current height taken up by the mixed layer, from 67% to 38% between the $(ReI,Sc)=(100,1)$ and (100, 10) cases [to only 17% in the $(ReI,Sc)=(100,100)$ case], and from 50% to 33% between $(ReI,Sc)=(500,1)$ and (500, 10). Dense fluid reaches closer to the front of the flow as Schmidt number is increased, with the percentage of the flow covered by the $ΔS̃=0.48$ contour increasing from 58% to 88% between the $(ReI,Sc)=(100,1)$ and $(ReI,Sc)=(100,10)$ cases [further increasing to 100% in the (100, 100) case] and from 80% to 88% between the $(ReI,Sc)=(500,1)$ and (500, 10) cases. This demonstrates that increasing Reynolds number reduces the influence of increased Schmidt number on some flow features, with increasing Schmidt number beyond ten having less impact than increasing from 1 to 10. Additionally, this Sc = 100 case has dense fluid reaching the very front of the flow. This suggests that a further increase to $Sc=O(1000)$ to match real-world solute-based flows would likely result in only minor changes in these parameters.

The effect of Reynolds number is more complex (see Fig. 6). While increasing Reynolds number consistently reduces current height (in this case, the decrease at Sc = 1 is proportional to $Re−0.5$⁠) and increases the percentage of flow length, the distance between inlet and flow front, covered by the $ΔS̃=0.48$ contour, the percentage of current height covered by the mixed layer increases in the Re_I = 3000 case compared with Re_I = 1000, perhaps a result of increased mixing by the Kelvin–Helmholtz structures that form with increased Reynolds number (see Fig. 7). These structures are also present in the Re_I = 1000 case and may be emerging in the $(ReI,Sc)=(500,10)$ case, in which the mixed layer thickness decreases as a proportion of current height compared with the lower Re_I cases. However, the rate of decrease between Re_I = 500 and Re_I = 1000 is slower than that between Re_I = 100 and Re_I = 500, which may also be a result of the Kelvin–Helmholtz structures.

The three-dimensional density isosurfaces (Fig. 7) indicate that these trends extend across the width of the tank. These isosurfaces show signs of structural change with increasing Reynolds and Schmidt numbers. At Re_I = 100, the isosurfaces from the cases with Sc = 1 and Sc = 10 are completely smooth. At Sc = 100, ridges appear in the highest density isosurface. In the $(ReI,Sc)=(1000,1)$ case, the lowest density isosurface has oscillations behind the head. By $(ReI,Sc)=(3000,1)$⁠, all isosurfaces have lost the smoothness of the $(ReI,Sc)=(100,1)$ case, as expected owing to the increase in turbulence. Therefore, depending on the flow Reynolds and Schmidt numbers, perturbations may be visible in the highest and/or lowest density isosurfaces. Additionally, with increasing Reynolds number the perturbations in the density isosurfaces are most pronounced near the head, while for increasing Schmidt number the perturbations become more pronounced with increasing distance from the head. This suggests that there are at least two distinct mechanisms influencing the flow structure.

To investigate the mechanisms responsible for these changes, pseudocolor plots of density both in an $X̃−Z̃$ plane at $Ỹ=0.1$ and in a $Z̃−Ỹ$ plane near the current front (Fig. 8) can be inspected. These plots show that cases $(ReI,Sc)=(100,100),(500,10)$⁠, and (3000, 1) contain regions of less-dense fluid surrounded by the denser fluid of the head. The $Z̃−Ỹ$ plane illustrates that this fluid is absorbed upward, originating from the over-running of ambient by the raised nose of the flow. Therefore, as well as causing dense fluid to reach closer to the front of the flow, increasing Schmidt number at fixed Reynolds number leads to the formation of lobe-and-cleft structures within the head. These structures also form with increasing Reynolds number.

In order to obtain a thorough understanding of the flow structure, the velocity structures are inspected. Pseudocolor plots of all three velocity components for each case are shown on an $X̃−Z̃$ slice close to the bottom boundary (Fig. 9) and on $X̃−Ỹ$ slices (Figs. 10 and 11). In addition to Figs. 7 and 8, Fig. 9 highlights a strong symmetry about the central $Z̃$-plane in all but the $(ReI,Sc)=(3000,1)$ case. Only in this case does the increase in nonlinearity owing to higher Reynolds number cause this symmetry to break. Excepting this highest Reynolds number case, the cross-stream velocity for all cases has negligible magnitude on the central $Z̃=0$ plane (Fig. 10), indicating a symmetric solution. However, away from this central plane, the magnitude of cross-stream and vertical velocities have equivalent magnitude in all cases (for example the plane at $Z̃=0.5$ shown in Fig. 11). This suggests that the flow is not two-dimensional as commonly assumed.^1,8 Downstream velocity on the two $X̃−Ỹ$ planes has the same structure and magnitude in all cases, as does cross-stream velocity in the Re_I = 3000 case.

The vertical velocity plots confirm the over-riding of ambient fluid, with Fig. 9 showing areas of large positive vertical velocity close to the bottom boundary and near the front of the current for the $(ReI,Sc)=(100,100)$⁠, (500, 10), and (3000, 1) cases, corresponding to the rising buoyant fluid over-ridden by the current front. Additionally, as expected from the density contours, increasing Reynolds number increases downstream velocity within the flow. Increasing Reynolds number also increases variation in all components of velocity within the body, including alternating positive and negative vertical velocity near the side walls in the $ReI=1000,3000$ cases. Visible in both Figs. 9 and 10, increasing Schmidt number results in a regular alternating pattern of positive and negative vertical velocity at a central cross-stream location, and corresponding positive and negative cross-stream velocity on either side of the center, behind the head of the current. The regular alternating vertical velocity pattern established by increasing Schmidt number is localized to the cross-stream center of the flow, though increasing Reynolds number increases the width of the motions (Figs. 9 and 11).

Figure 12 illustrates the flow behind the head in the $(ReI,Sc)=(500,1), (500,10)$ cases at $t̃=66.3$⁠, demonstrating that these are not short-term changes. As well as velocity plots, this figure contains plots of density fluctuations from cross-stream averaged density (⁠$ΔS̃−ΔS̃¯Z̃$⁠, where $ΔS̃¯Z̃$ is density averaged in the cross-stream direction). The density fluctuations in the Sc = 10 case contain a pattern of alternating positive and negative regions correlated with those in vertical velocity but with a $1/4$-wavelength offset (characteristic of internal gravity waves⁶¹). To understand why decreasing mass diffusivity leads to large-scale changes in flow structure, plots of swirling strength and gradient Richardson number will be inspected.

Figure 13 shows plots of swirling strength ζ_ci as defined by Zhou et al.,⁶² 

where L is the velocity gradient tensor, ζ_r and χ_r are the real eigenvalue and eigenvector, and $ζcr±iζci$ the complex conjugate pair of complex eigenvalues with corresponding eigenvectors $χcr±iχci$⁠. These plots reveal that increasing Schmidt number leads to the formation of structures in the mixed layer between the current and the ambient in the center of the tank in the cross-stream direction (with the mixed layer here defined as the region between the $ΔS̃=0.03$ and $ΔS̃=0.48$ contours). The placement of these structures is identical for the $(ReI,Sc)=(100,10),(100,100)$ cases, and their spacing in the downstream direction is the same for all three Sc >1 cases. Figure 13 shows that increasing Reynolds number also leads to the formation of structures within the body, though they differ from those resulting from increased Schmidt number. While some of the structures resulting from increased Reynolds number are within the mixed region, the structures in the higher Reynolds number cases have a less regular pattern (with a different physical spacing to the high Schmidt number cases) and are not limited to the center in the cross-stream direction.

Change in large-scale flow structures with Schmidt number is not immediately anticipated, given that Schmidt number is expected to result in changes at the small scale. Figure 14 shows the density and downstream velocity profiles on a central cross-stream location averaged over downstream location at the time step illustrated in Fig. 12. The change in diffusivity resulting from the Schmidt number increase leads to a change in density profile, specifically the anticipated increase in density in the lower part of the flow and sharper transition from dense to ambient fluid. As there is a greater density difference, there is a corresponding increase in downstream velocity within the body. These changes may affect the stability of the interface. To illustrate this, a gradient Richardson number can be calculated,

where ρ_c is the density of the fluid pumped in at the inlet, which gives a measure of the stability of density stratification. If Ri > 0.25, then the energy produced by shear is not sufficient to overcome the density stratification and is therefore dissipated.⁶³ Profiles of Ri for the cases with Re_I = 500 are shown in Fig. 14(c). Increasing Schmidt number from 1 changes the Ri profile such that the value at the current height moves from above to below this critical level. The same change is seen for every case with Sc >1, while for every case with Sc = 1, Ri > 0.25 in this area. Therefore, changes in the density and velocity profiles resulting from decreasing mass diffusivity (and therefore increasing Schmidt number) lead to the density stratification becoming less stable such that energy produced from shear is no longer dissipated but instead leads to large-scale structural changes in the flow.

Increasing the flow Reynolds number has been shown to result in a shorter head, with more velocity fluctuations, and greater front and internal velocities. Excepting the highest Reynolds number case considered in this work, a strong symmetry plane is present at a central cross-stream location for all cases. Attempting to quantify such flows using exclusively a central cross-stream plane could, depending on flow Reynolds number, give a misleading impression of the overall flow particularly in terms of the cross-stream velocity (Figs. 9–11). For most cases, W was found to be 0 on the central cross-stream plane but elsewhere the magnitude of cross-stream velocity was equivalent to that of vertical velocity suggesting that the flow is not two-dimensional as often assumed.^1,8

Figure 5 and Table III show that increasing either Schmidt or Reynolds number results in a more defined head, with a forward angled depression in the density contour behind the head. The head height, based on the $ΔS̃=0.03$ density contour, decreases slightly from $Ỹ≈1.2$ at Re_I = 100 to $Ỹ=1$ at Re_I = 500 with no further decrease when Reynolds number is increased further. Increasing Schmidt number does consistently reduce the head height, with a more significant change at lower Reynolds number and when Schmidt number is increased from 1 to 10 compared with 10 to 100. The difference between the rightmost positions of the $ΔS̃=0.03$ and $ΔS̃=0.48$ contours as a proportion of current front position decreases with both increased Reynolds and Schmidt numbers, indicating that dense fluid is reaching closer to the front of the flow. In the $(ReI, Sc)=(100, 100)$ case, the $ΔS̃=0.48$ contour reaches the front of the flow, suggesting that a further increase in Schmidt number would likely have little impact.

The lobe-and-cleft structures resulting from the over-running of ambient fluid by the current front are present in some cases but not others (Fig. 8). As lobe-and-cleft structures are associated with some of the largest bed shear stresses,^22,64 and changes in rates of mixing,⁶⁵ accurately capturing this feature is important to understanding the flow structure. While the $(ReI,Sc)=(100,1), (100,10)$⁠, and (500, 1) cases do not exhibit over-running of ambient fluid, the $(ReI,Sc)=(100,100)$ and (500, 10) cases do. Therefore, for Reynolds numbers $O(100)$ the presence of lobes-and-clefts in the current head is dependent on the Schmidt number [with Fig. 15(b) showing the phase space where these structures are found in this work]. This conflicts with the findings of Bonometti and Balachandar,³³ who suggest that lobe-and-cleft structures are not Schmidt number dependent. However, varying Schmidt number in three-dimensional DNS here was only possible at Reynolds numbers sufficiently low that these structures were not already present at Sc = 1. As over-running of ambient fluid is observed in the $(ReI,Sc)=(3000,1)$ case (Fig. 8), providing the Reynolds number of the flow is sufficiently high this flow feature may be captured without increasing Schmidt number above 1.

Considering flow behind the head, the current height decreases with both Reynolds and Schmidt numbers (Fig. 5 and Table III). However, the height of the $ΔS̃=0.48$ contour seems to be relatively consistent regardless of the Reynolds and Schmidt number of the flow (likely a result of the short duration of the simulations presented). Increasing Reynolds number from Re_I = 100 to Re_I = 500 leads to a smaller percentage decrease in current height as Schmidt number is increased. At the Reynolds and Schmidt number range considered in this work, the percentage change in thickness of the mixed layer of fluid behind the head as Schmidt number is increased from 1 to 10 is equivalent in both Re_I = 100 and Re_I = 500 cases. A further increase from Sc = 10 to Sc = 100, however, results in a smaller change. Increasing Reynolds number was found to have a more complex effect on mixed layer thickness, with the percentage of flow height taken up by the mixed layer initially decreasing and then increasing when Re_I = 3000. This may be a result of increased mixing from the Kelvin–Helmholtz structures (visible in Fig. 7) that begin to form as Re_I increases. Close examination of the density contours in Fig. 5 and the isosurfaces in Fig. 7 indicates that the Kelvin–Helmholtz structures may be emerging in the $(ReI,Sc)=(500,10)$ case [but not the $(ReI,Sc)=(500,1)$ case]. This suggests that increasing Schmidt number may reduce flow stability.

Several changes resulting from increased Schmidt number have been noted in the data from this paper. In many cases, the impact of increasing Schmidt number beyond one is either reduced by increasing Reynolds number (for example the change in current height), or the same changes are observed with increased Reynolds number (for example the presence of lobes-and-clefts). There are, however, features that are not captured if Sc = 1 is assumed. In particular, increasing Schmidt number is related to the formation of structures at the current–ambient interface behind the head. This can be seen in several of the plots presented, for example, the instability of the density contours (Fig. 5), in the velocity plots (Figs. 9 and 10), and in the swirling strength plots (Fig. 13), in which wave-like distortions in the density contours correlate with peaks in swirling strength.

The stability of the interface can be quantified by a gradient Richardson number in the upper part of the flow. In all cases with Sc = 1, even those with Kelvin–Helmholtz structures behind the head, the gradient Richardson number in the upper part of the flow is above the critical value. As discussed by Pelmard et al.,²⁴ a gradient Richardson number below 0.25 in the head may lead to the formation of Kelvin–Helmholtz structures that then dissipate some distance behind the head if the value rises above the critical level in the body. Therefore, the current–ambient interface in the body may be stable even with the presence of Kelvin–Helmholtz structures near the head. All cases with Sc >1 have Ri < 0.25 in the upper part of the flow, suggesting that density stratification is no longer stable enough to dissipate the energy generated through shear and exhibit the formation of corresponding structures on the current–ambient interface behind the head. These structures are not diminished as distance from the head increases (Fig. 13), suggesting that this may be a distinct mechanism from the formation of Kelvin–Helmholtz vortices, the influence of which decreases with distance from the head in these data, and which are present in some cases with Sc = 1 [see Fig. 15(a)]. A similar change from above to below the critical gradient Richardson number (with corresponding structure formation) was observed by Kneller et al.⁶⁶ in particulate flows, both with increasing bed slope and increasing particle settling velocity (or larger particle sizes).

The velocity and density perturbations associated with the structures in the cases with Ri < 0.25 (Fig. 12) have correlated patterns of alternating positive and negative regions, with the $1/4−$ wavelength offset characteristic of internal gravity waves,⁶¹ supporting the conclusions of Marshall et al.²⁵ The formation of these internal gravity waves is a result of the decrease in mass diffusivity sharpening the density profile, leading to a change in the stability of the interface. Crucially, unlike other characteristics, it does not appear to be the case that this effect of increased Schmidt number is diminished by increased Reynolds number in the range considered here. The perturbations in density field are at least as prominent in the Re_I = 500 case as in the Re_I = 100 cases (Fig. 5). The effect is also not captured purely by increasing Reynolds number in the range considered in this work. While peaks in swirling strength are found in the $ReI=1000, 3000$ cases, they are missing the regularity of those in the higher Sc cases and are not limited to the center in the cross-stream direction, supporting the suggestion that this is a separate mechanism to those seen with increased Reynolds number.

Many numerical investigations of gravity current flows have sought to mitigate the high computational cost of three-dimensional direct numerical simulation by claiming the effect of increasing Schmidt number from 1 is negligible. However, the justifiability of this assumption has been questionable given the lack of understanding regarding the effect of Schmidt number on three-dimensional flow features. In this work, the effects of Reynolds and Schmidt numbers on constant-influx solute-based gravity current flow structure have been investigated through three-dimensional direct numerical simulation performed using the spectral element solver Nek5000. These results have been used to draw conclusions regarding when a Sc = 1 assumption is justified.

The importance of considering Schmidt number is dependent on the flow property of interest and on the flow itself. Some flow features appear to be independent of Schmidt number (for example, current front velocity). Additionally, some of the effects of increased Schmidt number also occur with increased Reynolds number (such as the appearance of lobe-and-cleft structures in the head) or are reduced by increased Reynolds number (such as the change in current height). A notable exception is the reduction in gradient Richardson number. In flows with $ReI=100$ and 500, increasing Schmidt number from 1 to 10 was found to reduce the gradient Richardson number in the body of the flow from above to below the critical value, resulting in the presence of structures in the mixed layer. When moving from Re_I = 100 to Re_I = 500, this effect of increased Schmidt number was not reduced. Further, equivalent structures were not apparent in the Re_I = 3000 case, suggesting that this feature may not be captured purely by increasing Reynolds number. When considering the structure of the gravity current body in a high Schmidt number flow, assuming Sc = 1 may therefore lead to qualitative changes in flow structure (for example, to internal gravity waves). The importance of considering Schmidt number may also depend on flow type, for example, the data presented in this work suggest that Schmidt number impact may be greater in a more viscous flow (such as clay based transitional flows^67,68) though further work considering the impact of Reynolds number on such flows is needed.

A Schmidt number of 1 is often assumed when performing numerical investigations of gravity current flows. This is largely as a result of the rapidly escalating computational cost of DNS,^27,34 which scales with $Re3Sc2$⁠. When computational resources are limited, deciding whether to prioritize increasing Schmidt or Reynolds number is a complex issue dependent on several factors. There may be no benefit to prioritizing Schmidt number at the expense of Reynolds number if data analysis will focus on parameters that are not Schmidt number dependent (such as front velocity), or that are also seen with increased Reynolds number (such as the formation of lobe-and-cleft structures). However, it is recommended to have Schmidt number sufficiently large to accurately capture the body gradient Richardson number. The work presented here suggests that assuming a Schmidt number of 1 in numerical investigations leads to substantial structural differences compared with higher Schmidt number flows. However, even at Re_I = 100 the effect of increasing Schmidt number from 10 to 100 was quantitative rather than qualitative [similar to the findings of Deepwell and Stastna⁴⁷ for flow along a pycnocline(s)], and therefore the structure of solute-based flows [$Sc=O(1000)$] can likely be captured with the comparatively minor cost of a small increase in Schmidt number rather than the large cost of matching Schmidt number exactly.

We thank the reviewers and editor for their feedback on this paper. The numerical simulations presented were carried out on the ARC3 and ARC4 supercomputing facilities at the University of Leeds. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Fluid Dynamics at the University of Leeds, Grant No. EP/L01615X/1. R.M.D. was supported by the NERC Grant No. NE/S014535/1. S.D. would like to acknowledge the computer time provided by NCSA, UIUC on the Blue Waters supercomputer at NCSA, which was used for the initial setup and testing of the numerical simulations.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.