Numerical validation of novel scaling laws for air entrainment in water

The Froude scaling laws have been used to model a wide range of water flows at reduced size for almost a century. In such Froude scale models, significant scale effects for air–water flows (e.g. hydraulic jumps or wave breaking) are typically observed. This study introduces novel scaling laws, excluding scale effects in the modelling of air–water flows. This is achieved by deriving the conditions under which the governing equations are self-similar. The one-parameter Lie group of point-scaling transformations is applied to the Reynolds-averaged Navier–Stokes equations, including surface tension effects. The scaling relationships between variables are derived for the flow variables, fluid properties and initial and boundary conditions. Numerical simulations are conducted to validate the novel scaling laws for (i) a dam break flow interacting with an obstacle and (ii) a vertical plunging water jet. Results for flow variables, void fraction and turbulent kinetic energy are shown to be self-similar at different scales, i.e. they collapse in dimensionless form. Moreover, these results are compared with those obtained using the traditional Froude scaling laws, showing significant scale effects. The novel scaling laws are a more universal and flexible alternative with a genuine potential to improve laboratory modelling of air–water flows.


Introduction
Physical modelling at reduced size is one of the oldest and most important design tools in hydraulic engineering. For processes of engineering interest involving free surface flows, the Froude scaling laws have been used since they were introduced by Moritz Weber in 1930 [1]. They ensure that the ratio between transformations. This article shows, by simulating a range of scales, that the derived selfsimilar conditions for air-water flows and their boundary conditions can be used to achieve self-similarity.
The derived scaling laws are applied numerically to two air-water flow processes, namely (i) a dam break flow interacting with an obstacle, generating large deformations of the free surface [34], and (ii) a vertical circular plunging water jet impinging on quiescent water characterized by significant air entrainment, based on the experimental results of [13]. The 3D RANS equations govern phenomena in which both viscosity and surface tension play a central role.
Air-water flows are here simulated by using interFoam, a numerical solver for two-phase incompressible fluid flows based on the volume of fluid (VOF) method implemented in the OpenFOAM v.1706 CFD package [35]. In these simulations, all the boundary and initial conditions, including the properties of the fluid, are transformed using the novel scaling laws at different geometrical scales with scale factors λ = l p /l m , where l p is a characteristic length in the prototype (subscript p) and l m is the corresponding one in the model (subscript m). The processes are scaled using values of λ for which a correct representation of surface tension and viscous effects is essential to avoid significant scale effects. The two processes are also simulated with the commonly applied Froude scaling laws using ordinary water and air in the model, as is common in laboratory experiments (herein called traditional Froude scaling), and the Froude scaling laws in which the properties of the fluids are strictly scaled (herein called precise Froude scaling). It is demonstrated that the novel scaling laws involve no scale effects, in contrast to traditional Froude scaling, and they are also more universal and flexible than precise Froude scaling.
This article is organized as follows: in §2, the Lie group transformations are applied to the governing equations and the novel scaling laws are derived. The numerical model is presented in §3. Subsequently, the two CFD case studies are illustrated in §4, including the set-up, the application of the novel scaling laws and the results. The findings of this research are discussed in §5 and the conclusions and recommendations for future work are given in §6. Finally, appendix A includes the details of the derivation of the novel scaling laws and the self-similar conditions owing to the initial and boundary conditions.

Analytical derivation of the novel scaling laws (a) Governing equations
Air-water flows are here described by the RANS equations for incompressible fluids and where i is the free index, j is the dummy index, following Einstein's notation, t is time, x i and x j are the spatial coordinates, U i and U j are the Reynolds-averaged flow velocity components, u i and u j are the fluctuating velocity components, u i u j is the Reynolds stress term, p is the Reynoldsaveraged pressure, ν is the kinematic viscosity, ρ is the density of the fluid, g i = (g 1 , g 2 , g 3 ) is the gravitational acceleration vector and f σ is the surface tension force per unit volume, defined as In equation (2.3), σ is the surface tension constant, κ is the curvature of the free surface and γ is the phase fraction. This is a dimensionless variable with values between 0 and 1 that is used to identify any air-water interface (see §3). The k-model is here applied for the Reynolds stresses in equation (2.2) (see [36] for more details), for which (2.4) where k is the turbulent kinetic energy, δ ij is the Kronecker delta and ν t is the eddy viscosity, (2. 5) k and its rate of dissipation are calculated from (2.6) and (2.7) P k = u t (∂U i /∂x j )(∂U i /∂x j + ∂U j /∂x i ) and C 1 = 1.44, C 2 = 1.92, C μ = 0.09, C σ k = 1.0 and C σ = 1.3 are the standard model coefficients used in the k-turbulence model [37]. The approach used here for turbulence modelling, combined with the VOF method, has been recognized to overestimate k [38], although this does not affect the derivation of the scaling laws and the self-similarity of the representation of the process.

(b) One-parameter Lie group transformations
The Lie group is defined as φ = β α φ φ * . (2.8) Equation (2.8) transforms the variable φ in the original space into the variable φ * in the transformed (*) space, β is the scaling parameter and α φ is the scaling exponent of the variable φ.
which, with β being a constant parameter, is rearranged as Self-similarity is achieved if equation (2.11) can be obtained from equation (2.1) by means of a simple scaling process. Therefore, all terms of equation (2.11) must be transformed by using the same scaling ratios To attain self-similarity of air-water flows, the exponents for length, velocity and fluctuating velocity components have to be identical for the ith axis. This is shown in appendix A, where the detailed derivation of equations (2.2)-(2.7) is presented. Hereafter, α x , α U and α u are used to indicate the scaling exponents of length, Reynolds-averaged velocity and fluctuating velocity components on the ith axis. Similarly, α ρ , α ν and α σ are derived by applying the Lie group transformations to equation (2.3). Furthermore, based on equations (2.5)-(2.7), the scaling conditions for the turbulent parameters are derived. In addition, the detailed derivations of the self-similar conditions for the initial and boundary conditions are also shown in appendix A. The scaling conditions derived above are summarized in the second column of table 1. They are consistent with those reported in tables 1 and 2 in [29] with addition of the surface tension and the curvature of the free surface. All the exponents are written in terms of three independent scaling exponents, namely α x , α t and α ρ , meaning that they are user defined (their choice is flexible). In fact, the solution of air-water flow equations can be mapped to solutions in other transformed domains with different λ = β α x by selecting the scaling parameter β and changing the α of three independent variables.
It is possible to assign the value of one or two of the three α while still preserving self-similarity. For example, in table 1, it is shown that choosing α g = 0 implies that α g = α x − 2α t . Therefore, the unscaled g requires that α t = 0.5α x . In this configuration, the remaining scaling exponents are written in terms of α x , α ρ and α g = 0 (fourth and fifth columns of table 1). Hence, keeping g invariant in a scaled model requires the time and flow velocities to be scaled and the properties of the fluids to be changed to obtain a self-similar behaviour. A further restriction can be imposed on the density of the fluids, namely α ρ = 0. This restriction leads to the well-known precise Froude scaling laws [3], as a particular case of the novel scaling laws, where g is constant and ν and ρ are scaled by keeping Re and We invariant.

Numerical model
Air-water flows are simulated by using the two-phase flow solver interFoam, based on the VOF method, implemented in the OpenFOAM v1706 CFD package [35]. A single system of RANS equations is solved with the pressure and velocity fields shared among both phases. The interface between water and air is identified by a value of the phase fraction γ between γ = 1 (water) and γ = 0 (air). The fluid properties used in the equations are mapped in all domains as a weighted average using γ as weight, e.g. for ρ and ν ρ = γρ w + (1 − γ )ρ a (3.1) and where subscripts w and a refer to the water and air phase, respectively. σ appears in equation (2.3) to model the surface tension force per unit volume, as stated in the continuum surface force method proposed by [40]. The curvature of the interface between two fluids κ is defined as γ is transported as a scalar by the flow field and the interface location (e.g. the free surface) is updated by solving the volume fraction equation The interface reconstruction technique used by interFoam is MULES [41]. The free surface can also be captured by using alternative techniques, such as the isoAdvector method [42]. However, the governing equations remain the same and the self-similarity of the representation of the process under the novel scaling laws is not affected by the interface reconstruction technique.              x 3

Numerical results
The self-similar conditions of the novel scaling laws are validated with the simulation of two physical processes: (i) a dam break flow interacting with an obstacle and (ii) a vertical plunging water jet. The simulations for both processes involve the prototype and a number of scaled models up to large geometrical scale factors of λ = 16.

(a) Dam break flow
Dam break flows have been widely investigated numerically and the specific case addressed herein is chosen because it is a well-known test to validate the modelling of large deformations of free surfaces [43,44]. The solver used in the present study has been validated with this particular test case by [34]. In this study, γ = 0.1 is selected to identify the air-water interface in the VOF method. γ = 0.1 is obtained by considering the value between 0 and 1 providing the best fit with the experimental void fraction distribution in §4b.

(i) Numerical set-up
The initial condition at t = 0 consists of a quiescent water column of volume 1.228 × 0.550 × 1.000 m 3 , located at the left side of a 3.220 × 1.000 × 1.000 m 3 tank (figure 1). A prismatic fixed obstacle with a volume of 0.160 × 0.160 × 0.403 m 3 is located at x 1 = 2.395 m. The water column is released instantaneously at t = 0. Subsequently, the flow impacts the obstacle and creates a complex two-phase flow. The top wall of the domain is modelled as an open, fully transmissive boundary at atmospheric pressure and all the remaining walls as no-slip boundary conditions. The water density is ρ w = 1000 kg m −3 , its kinematic viscosity is ν w = 1 × 10 −6 m 2 s −1 and the surface tension constant is σ = 0.07 N m −1 .
A 180 (length) × 60 (width) × 80 (height) Cartesian computational grid was used, apart from the obstacle. Note that, because of the orientation of the reference frame, for this case g i = (0, 0, −g) in equation ( where x j is the mesh size in the Cartesian coordinate system and C max = 0.8 is the maximum Courant number following [45]. The simulations were run on the University of Nottingham highperformance computing (HPC) cluster Augusta. The number of cells in the computational domain was 861 075 and the used cores and memory were 4 and 36 GB, respectively. It required 2 h to simulate the actual time of 6 s (also for the corresponding times at reduced scales). In this test case, as well as for the jet, all the dimensional parameters, including the mesh sizes and time steps, were scaled to the smaller domains according to the selected scaling laws.
(ii) Application of the novel scaling laws Two self-similar domains, namely D8 and D16, are created with geometrical scale factors of λ = β α x = 8 and 16, respectively. To achieve this, it is assumed that α x = 1, such that β = 8 (D8) and 16 (D16), respectively. All variables and parameters are transformed by the scaling exponents in the fourth and fifth columns of table 2 (with scaling conditions in terms of α x , α t and α g = 0). Their specific values for the prototype and the scaled models, obtained by applying the conditions in

(iii) Results
For the purpose of this work, it is interesting to analyse the time when gravity, inertial, viscous and surface tension effects are all relevant. This happens when the dam break flow impacts the obstacle and creates an elongated water tongue. Figure 2 shows this process with snapshots of the prototype and the scaled domains at x 2 = x 2 /h w = 0 (figure 1) and dimensionless time t = t g/h w = 2.7. The contours in figure 2 represent the dimensionless velocity magnitude U = U/ gh w , where U = U 2 1 + U 2 2 + U 2 3 . The prototype shows a large free surface deformation after impacting the obstacle (figure 2a). The self-similar domains and the domains scaled with precise Froude scaling all simulate the water tongue of the prototype correctly. Moreover, the dimensionless velocity magnitudes in the prototype and in the self-similar domains are the same, despite the increasing λ (figure 2b-e). On the other hand, traditional Froude scaling does not model the free surface correctly owing to Re and We scale effects, i.e. the water tongue becomes less prolonged with increasing λ (figure 2f,g).
The differences between the prototype and the scaled domains are quantified using the root mean square error along the plane where U b,p are the cell values of U in the prototype, U b,m are the scaled domains and n = 14283 is the number of cells in the cross-section x 2 = 0. As shown in table 4, the RMSE U values for D8 and D16 confirm a nearly perfect self-similarity with respect to the prototype. k is used to assess turbulence because it shows significant scale effects if ν is not scaled. Air entrainment is assessed by using γ , which is expected to deviate from the prototype if the surface tension is over-represented in the scaled domain. Figure 3 shows     The perfect collapse of the data for D1, D8 and D16 affirms the self-similar behaviour of k for the novel scaling laws. The self-similar behaviour is also confirmed for D8 PFr and D16 PFr . On the other hand, k shows scale effects using traditional Froude scaling; the first k peak is either under-or overestimated (D8 TFr and D16 TFr , respectively), while the magnitude of the second peak decreases with increasing λ.
As demonstrated in figure 4, where γ is shown as a proxy for surface tension, air entrainment is correctly scaled in the self-similar domains as it controls the air-water interface and the free surface curvature. While the results in the domains D1, D8, D16, D8 PFr and D16 PFr essentially collapse, the domains scaled with traditional Froude scaling show significant differences in the region where air entrainment is most important. γ starts to increase close to t = 4, meaning that the wave reaches RW consistently at the same time in all domains except for D8 TFr and D16 TFr ( figure 4). Subsequently, γ increases to reach 1 less rapidly than in the prototype when using traditional Froude scaling. These differences become more visible at a later stage of the simulation when the dam break wave is re-reflected at t = 23.6, showing significant scale effects.

(b) Plunging water jet
In this section, the same scaling laws as in the previous test case are applied to the plunging water jet presented in [13]. This involves free-surface instabilities, air entrainment and turbulence.

(i) Numerical set-up
The set-up is based on the experiments of [13], consisting of a jet from a circular orifice impinging on a prismatic column of water. However, in this study, the symmetry of the problem with respect to two orthogonal vertical planes is used to simulate only a quarter of the domain, in order to reduce the computational cost. Figure 5 shows the numerical domain and the variables used in the prototype. A plunging water jet is ejected from a nozzle having a radius r in = 0.0125 m. Here, the subscript in indicates the quantities at the nozzle, i.e. at the inlet of the numerical domain, while the subscript im indicates values of variables at the still water level, i.e. x 1 = 0 . The receiving pool is 0.15 m wide and 1.80 m deep and at the start of the simulation the distance between the water surface and the nozzle is l 1 = 0.10 m. The velocity of the jet at x 1 = 0 is U im = 4.10 m s −1 . Here, a Cartesian coordinate system with x 1 pointing downwards is used; therefore, g i = (g, 0, 0).
The inlet boundary condition, namely the nozzle, is at the top boundary. The velocity at the inlet U in and both k in and in are prescribed, while the outlet is located at the bottom boundary, having the same flow rate magnitude as the inlet.
U in is calculated starting from the jet impact velocity using Bernoulli's theorem U in = where I = 0.46% is the turbulence intensity following [13], and l t is the turbulence mixing length approximated with l t = 0.07r in . The part of the top boundary of the domain not occupied by the inlet was modelled as a fully transmissive open boundary at atmospheric pressure. Since only a quarter of the domain is simulated, a symmetry boundary condition is used at the symmetry boundary walls and no-slip conditions are applied at the remaining walls, including the bottom wall outside the outlet cells ( figure 5).
A structured orthogonal mesh is used with a finer resolution for the area in which the water jet impacts the free surface down to a depth of 0.6 m. The smallest observed bubble size was 1 mm and the minimum cell size was 0.625 mm to increase the interface sharpness around the bubbles [13,46]. This mesh resolution is not fine enough to resolve the smallest bubbles present in the flow. However, the main focus of this work is to show the relative differences in the results of the application of different scaling laws for air-water flows, rather than to perfectly resolve the dynamics of individual bubbles. The simulation time was 300 s, the same duration as that used by [13] to compute the distribution of the void fraction from the laboratory measurements, and the time step varied with respect to the CFL condition. C max was set equal to 0.3. The simulations were run on the University of Nottingham HPC cluster Augusta. The number of cells in the computational domain was 1.89 × 10 6 and the corresponding cores and memory were 10 and 36 GB, respectively. It required 168 h to simulate 300 s actual time (also for the corresponding times at reduced scales).

(ii) Application of the novel scaling laws
The two self-similar domains P8 and P16 were simulated with geometrical scale factors of λ = 8 and 16, respectively. Similarly to the dam break case, the scaling exponent for length is α x = 1 so that β = 8 (P8) and 16 (P16). The scaling ratios and parameters obtained by applying the conditions in the second column of    the description of high Re plunging jets provided by [47]. In particular, the flow shows the characteristic conical shape of the air-entrainment layer and the dispersion of bubbles due to the buoyancy effects outside the cone. The consequence of air entrainment in the flow is a rise of the free surface with respect to the initial conditions (figure 6a-c). Domains P8 and P8 PFr have an identical shape to the air-entrainment layer, showing that the free surface reaches the same level, while P8 TFr shows clear differences.

(iii) Results
The following results are all shown along section A-A' at (x 1 − l 1 )/r im = 1.60. The distribution of the void fraction is compared with the experimental results of [13] in figure 7. The computed distribution and that measured in [13] are shown to have a close agreement. The novel scaling laws and precise Froude scaling reproduce the distribution of the void fraction of the prototype correctly, in terms of both the shape and magnitude. On the other hand, the traditional Froude scaling fails to describe the void fraction distribution. Figure 8 shows the time-averaged dimensionless velocity magnitude U , where for this case U = U/U im . In the prototype, the maximum value of U is at the jet centreline and U follows qualitatively the same velocity distribution as found in [48]. While the results of the domains P1, P8, P16, P8 PFr and P16 PFr are identical, U for the domains P8 TFr and P16 TFr are lower than in the prototype. Figure 9 shows the time-averaged dimensionless turbulent kinetic energy k , where k = k/(gr im ). In the prototype and self-similar domains the maximum value is k = 10 at s/r im = 1. peak and remains almost constant as far as s/r im = 1.0, beyond which it decreases. Moreover, the value of k around the jet is higher in P8 TFr than in the prototype. However, P16 TFr shows a lower k than the prototype with a maximum value of k ≈ 4.5.

Discussion
Self-similarity has been achieved for the governing equations of air-water flows, including surface tension expanding the scaling conditions reported in [28,29]. An advantage of this approach is that the scaling conditions are directly derived from the governing equations. This leads to more universal scaling laws than the Froude scaling laws [49]. Furthermore, the choice of the scaling exponents α x , α t and α ρ in the second column of table 1 are user defined (flexible). This implies that novel scaling laws can also be written in terms of a set of other variables to find different configurations. For example, it is shown that precise Froude scaling is obtained as a special case of the novel scaling laws. The CFD simulations conducted herein demonstrated that both the novel scaling laws and precise Froude scaling result in self-similar air-water flows, which would also be the case for another set of variables.
In the dam break flow, a significant deformation of the free surface is shown in the prototype after the flow impacts the obstacle, with a characteristic water tongue projected downstream of the obstacle. This behaviour is captured in all the domains scaled with the novel scaling laws; figure 3 show that k is the same by using the novel scaling laws and k is thus self-similar. The phase fraction is also self-similar. This is a strong indication that surface tension effects are self-similar as well (figure 4) and it is also true for the domains D8 PFr and D16 PFr , since precise Froude scaling is a special case of the novel scaling laws. On the other hand, the commonly applied traditional Froude scaling, relying on the same fluids as in the prototype, fails to reproduce the behaviour of the prototype. Indeed, figure 2f,g shows that the water tongue is not well predicted. After t = 2.7, it collapses and the flow is reflected at the downstream wall. Scale effects are observed in k and γ at point RW. Furthermore, the flow reaches point RW later than in the prototype with increasing λ. Scale effects are also observed after the flow is re-reflected, particularly at the second peak of k .
For the plunging jet, air entrainment plays a central role. Figures 8a and 9a demonstrate that the novel scaling laws result in self-similarity for U and k , i.e. these results collapse for P1, P8, P16, P8 PFr and P16 PFr , while this is not the case for P8 TFr and P16 TFr . The self-similarity of the distribution of the void fraction depends on density, viscosity and surface tension effects. The prototype simulation captures the mechanism of air entrainment by a plunging jet (figure 5), including the formation of an air cavity between the impinging jet and the surrounding fluid, which collapses and reforms intermittently, entraining air bubbles that are transported by the flow. At this stage, air bubbles are advected in a turbulent shear flow and they are broken into smaller bubbles, creating a conical air-entrainment layer. Subsequently, buoyancy determines the re-surfacing of bubbles in the portion of the flow outside the air layer [7,8,12]. This complex mechanism causes the air-entrainment layer in figure 6, where the novel scaling laws guarantee self-similarity. This is also true for the void fraction in figure 7, which is a consequence of the mechanism described above. On the other hand, figure 7b demonstrates that traditional Froude scaling fails to reproduce the void fraction distribution. By using ordinary water, the surface tension and viscosity are over-represented; therefore, the distribution of the void fraction gradually decreases with increasing λ. As expected, for increasing λ the flow regime changes, transitioning from high Re = 50 840 in the prototype to Re = 800 in P16 TFr , calculated by using U im , r im and ν w . The use of k-, in this case, introduces also model, in addition to scale effects [2,3,47], which explain the results in figures 8b and 9b.
The need for novel scaling laws for scaling fluid properties requires the modification or replacement of ordinary water in laboratory experiments, e.g. for values of λ comparable to the highest used here, i.e. λ = 16, where ρ w = 62.5 kg m −3 , ν w = 1.56 × 10 −8 m 2 s −1 and σ = 1.70 × 10 −5 N m −1 (tables 3 and 5). There are options to alter the relevant fluid properties: the surface tension can be modified by adding ethanol to water [11] and the viscosity can also be reduced, e.g. Rouse et al. [50] modelled a hydraulic jump with air. A more recent approach to change the water properties is based on nanofluids, i.e. nanoparticles are added to water [51,52]. A key advantage of the novel scaling laws is that fluids of different density from water, e.g. ethanol, now also qualify as potential candidates for laboratory experiments.

Conclusion
The Froude scaling laws have been applied to model water flows at reduced size for almost a century. A significant disadvantage of Froude scaling is the potential for scale effects. This article shows how such scale effects in air-water flows are avoided with novel scaling laws based upon self-similarity of the governing equations. Lie group transformations are applied to the Reynoldsaveraged Navier-Stokes equations where surface tension effects are included as a source term. This allows the modelling of hydrodynamic phenomena at small scale without viscous and surface tension scale effects. These novel scaling laws are more universal and flexible than the precise Froude scaling laws because different scaling configurations can be obtained, e.g. by also scaling the density of the fluid. In this study, the gravitational acceleration is kept constant and the scaling exponents of the variables are expressed as a function of the scaling exponents of the length α x , time α t and gravitational acceleration α g = 0.
The derived novel scaling laws were validated with the simulations of two air-water flow phenomena: (i) a dam break flow interacting with an obstacle and (ii) a plunging water jet. The numerical simulations demonstrated that the processes are correctly scaled, and showed perfect agreement at different scales for air entrainment and kinematic properties. The results of the precise Froude scaling, where the properties of the fluids are strictly scaled, demonstrate that a particular configuration of the novel scaling laws is also able to result in self-similarity. By contrast, the simulations based on traditional Froude scaling using ordinary water and air, as is common in laboratory studies, show significant scale effects, as expected.
While this study provides a thorough numerical validation of the proposed scaling laws, future work aims to identify suitable fluids satisfying the novel scaling laws, which would enable the scaling of air-water flows without scale effects for the first time in a laboratory environment.
Data accessibility. All the OpenFOAM set-ups used for this article are available via Dryad: https://datadryad. org/stash/share/nY73jiRnKRNqgSsMnX2J3V4fSsywxEfGxUeZSXjyn1k.
Competing interests. We declare we have no competing interests.
Self-similarity is guaranteed if the scaling ratios of all terms in equation (A 1) are the same, implying that the exponents of all terms must be the same The Lie group transformations for equation (2.3) result in The dimension κ is the inverse of a length such that α κ = −α x i . Furthermore, α γ = 0 because γ is dimensionless. Hence, equation (A 5) reduces to From equations (A 2)-(A 4), the scaling exponents of the length dimensions along the ith axis are obtained as In other words, the scaling exponents of the length scale must be identical for i = 1, 2, 3 because the fluids are considered isotropic; therefore,