The implementation in OpenFOAM is preceded by an extension of the thermophysical model library, which is used inter alia by the solver compressibleInterFoam. On this note, a short explanation of the numerical application is given.

CompressibleInterFoam is a solver for two compressible, immiscible, non-isothermal phases (liquids and/or gases), whereby the interface is captured by the volume of fluid approach [13]. Consistently the Navier-Stokes equations (see Equations (6)-(9)) are solved for one fluid, where the considered fluid parameters are related to the phase distribution within the cell. The volume fraction of the liquid is represented by α , accordingly α = 1 corresponds to a cell completely filled with liquid and α = 0 to a cell full of gas. Based on the assumption of a homogeneous mixture, the cell specific density and viscosity ρ and μ are calculated by

ρ = α ⋅ ρ l i q u i d + ( 1 − α ) ⋅ ρ g a s , (1)

μ = α ⋅ μ l i q u i d + ( 1 − α ) ⋅ μ g a s . (2)

To track the interface of the two compressible phases, a transport equation for the volume fraction related to the fluid velocity vector U is used.

∂ α ∂ t + ∇ ⋅ ( U α ) + ∇ ⋅ ( U r e l ⋅ α ( 1 − α ) ) = α ⋅ ( 1 − α ) ⋅ ( ψ g a s p g a s − ψ l i q u i d p l i q u i d ) ⋅ D p D t + α ∇ ⋅ U (3)

The third term on the left-hand side of Equation (3) was created to sharpen the interface and avoid numerical diffusion. Details regarding the relative velocity vector U r e l and the numerical implementation can be found in [14]. The right-hand side takes into account the pressure p and its influence on the densities of both phases in relation to their specific compressibility ψ [15]. Considering the fluid temperature T and the thermophysical behavior of liquids and gases [16], the equations of states can be obtained.

ρ l i q u i d = ψ l i q u i d ⋅ p + ρ 0 l i q u i d = 1 R l i q u i d ⋅ T ⋅ p + ρ 0 l i q u i d (4)

ρ g a s = ψ g a s ⋅ p = 1 R g a s ⋅ T ⋅ p (5)

In terms of comparability, all presented studies have been carried out with general properties related to air and water. Hence, the gas constant for air ( R g a s = 287   J ⋅ kg − 1 ⋅ K − 1 [17] ), and common properties for water ( R l i q u i d = 3000   J ⋅ kg − 1 ⋅ K − 1 , ρ 0 l i q u i d = 1000   kg ⋅ m − 3 [18] ) have been used.

Based on the calculated mixture viscosity (see Equation (1)), the total mass continuity equation can be written as

∂ ρ ∂ t + ∇ ⋅ ρ U = 0 . (6)

Furthermore, the single momentum equation, including the dynamic viscosity μ , the unit tensor I and the gravitational acceleration g is defined as stated in Equation (7).

∂ ρ U ∂ t + ∇ ⋅ ( ρ U U ) = − ∇ p + ∇ ⋅ [ μ ( ( ∇ U + ( ∇ U ) T ) − 2 3 ( ∇ ⋅ U ) I ) ] − ∇ ρ g ⋅ x         + ∫ S σ κ ( x ` ) n ( x ` ) δ ( x − x ` ) d S (7)

The surface integral with the constant surface tension σ denotes the force acting at the liquid-gas interface [19], where N determines the unit normal to the interface and κ twice the mean curvature of the interface. The Dirac delta in three dimensions is expressed by δ ( x − x ` ) , containing x ` , a point on the surface and x , the point where the equation is calculated [20].

Finally, the energy equation applied in compressibleInterFoam [21] is stated in Equation (8).

∂ ρ T ∂ t + ∇ ⋅ ( ρ U T ) − ∇ ⋅ ( β ∇ T ) = − ( α c v , l i q u i d + 1 − α c v , g a s ) ⋅ ( ∂ ρ k 1 ∂ t + ∇ ⋅ ( ρ U k 1 ) + ∇ ⋅ ( U p ) ) (8)

The kinetic energy is expressed by k 1 = 0.5 ⋅ | U | 2 and the thermal diffusivity of the mixture by β . The specific isochoric heat capacity of the liquid and the gas phase is represented by c v , l i q u i d and c v , g a s , respectively, where values of 4182 J∙kg⁻¹∙K⁻¹ and 1007 J∙kg⁻¹∙K⁻¹ [17] have been used for each simulation shown in this paper.

Many different strain rate based viscosity models are existing for non-Newtonian fluids. Therefore, the five most common ones have been implemented within this work and are presented in the following section, whereas the strain rate γ ˙ is calculated in the same way as stated in the strainRateFunction of OpenFOAM [22].

γ ˙ = 2 ⋅ | 1 2 ( ∇ U + ( ∇ U ) T ) | (9)

One of the first developed non-Newtonian model is the well-known two parameter equation called Power Law, published by Reiner in 1926 [23]. Beside the strain rate, the fluid specific flow consistency index k 2 and the flow behavior index n are used to calculate the viscosity field.

μ = k 2 γ ˙ n − 1 (10)

In the very same year, Herschel and Bulkley propagated their widely used minima function model [24], considering a minimal viscosity μ 0 and a threshold strain stress τ 0 combined with the Power Law equation.

μ = min ( μ 0 , τ 0 γ ˙ + k 2 γ ˙ n − 1 ) (11)

However, the model presented by Casson in 1959 [25] consists only of a threshold strain stress and one additional fluid specific parameter m .

μ = ( τ 0 γ ˙ + m ) 2 (12)

In contrast, the Cross Power Law model developed in 1965 [26] contains a minimum and a maximum viscosity μ 0 and μ ∞ , and the two rheological parameters k 3 and n .

μ = μ ∞ + μ 0 − μ ∞ 1 + ( k 3 γ ˙ ) n (13)

The Bird-Carreau equation published in 1972 is also expressed by a minimum and a maximum value as well as by two parameters k 3 and n [27].

μ = μ ∞ + ( μ 0 − μ ∞ ) ⋅ ( 1 + ( k 3 γ ˙ ) 2 ) n − 1 2 (14)

The above stated non-Newtonian transport models were implemented in the thermophysical model library of OpenFOAM 5.x. To this end, the major implementation steps are briefly explained.

First, the appropriate file and folder structure was created for each model and the header files are included in psiThermos.C and rhoThermos.C, respectively. Up to now, only temperature and pressure-based viscosity models were available in the thermophysical model library of OpenFOAM. Hence, the current velocity vector has to be accessed within the viscosity calculation loop according to Listing 1. Subsequently, a scalar field for the strain rate can be implemented in both heRhoThermo.C and hePsiThermo.C and derived in relation to Equation (9) as written in Listing 2. In order to pass the strain rate to each model specific viscosity calculation function, e.g. CrosspowerlawTransortI.H, the transferred parameters had to be extended as shown in Listing 3. Furthermore, the specific viscosity computation is implemented for each model. To avoid calculation errors, a strain rate of 0 s⁻¹ is replaced by a value of 1E-10 s⁻¹. Related to Equation (13), Listing 4 exemplarily shows the implementation of the Cross Power Law model in OpenFOAM. To ensure a pleasant input option for the characteristic fluid properties, the standard structure of the software is maintained so that the model-specific parameters can be entered in the respective file in the constant folder, e.g. thermophysicalProperties.water. For this purpose, appropriate variables need to be added to the respective header files and initialized in the c files. Listing 5 illustrates the initialization for the four new parameters related to the

Listing 1. OpenFOAM: Accessing the current velocity field.

Listing 2. OpenFOAM: Computation of the strain rate field.

Listing 3. OpenFOAM: Extension of the passed variables by the strain rate components.

Listing 4. OpenFOAM: Computation of the Cross Power Law transport model.

Listing 5. OpenFOAM: Initialization of the new fluid parameters in CrosspowerlawTransport.C.

Cross Power Law. Finally, the storage of all relevant fields in respect to the defined output conditions enables a comprehensive analysis of the simulation results. As an example, the appropriate source code for the strain rate is provided in Listing 6.

To verify the implementations, pressure driven test simulations for each viscosity model have been carried out. The used computational domain is a three-dimensional horizontal backward facing step as depicted in Figure 1. The mesh consists of 246,800 hexahedral elements with a maximum edge length of 0.85 mm. In combination with a smooth transition to a minimum grid spacing of 0.2 mm at all walls and close to the step area with no hanging nodes, an acceptable numerical accuracy is achieved. Since the focus within this work is on the validation of the novel implementations, a grid study, as well as further optimization of the numerical solution properties has been neglected.

At the initial time t = 0 the whole geometry is filled with the respective liquid, and air is entering the domain with a constant viscosity of 1.8E−5 Pa∙s [17]. The examined non-Newtonian liquids have been chosen from literature in relation to the implemented models. Their rheological properties are listed in Table 1. In the interests of comparability, a simulation with water, using the already implemented const transport model, has also been conducted.

All other properties of the numerical setup remain the same for all simulations. Concerning this, Table 2 shows the boundary conditions including the

Listing 6. OpenFOAM: Storage of the strain rate field in heRhoThermo.C.