1 Introduction

Recent developments of orbital manoeuvring systems and upper rocket stages, such as the cryogenic Ariane 6 Vinci engine, aim to replace the conventional propellant hydrazine with environmentally more friendly and operationally safer substitutes. Such a substitute can be a conventional fuel such as hydrogen, methane or kerosene. The replacement of hypergolic with conventional fuels requires a better understanding of the process of injection and ignition under the extreme conditions that prevail in space. Due to the near vacuum conditions in space, the typically cryogenic fuel and oxidizer are injected into a superheated state, which leads to a rapid and strong evaporation. This process of nearly instant evaporation of the liquid is called flash boiling or flash evaporation. Understanding this process and subsequent mixing of the two reactants is paramount to develop a stable and reliable ignition. Therefore, a holistic approach is needed to numerically and experimentally investigate this process since bubble nucleation at a nanometer scale, growth, breakup, coalescence at a macroscale and lastly the evaporation of droplets at a micrometer scale impose a wide range of scales that cannot easily be covered by conventional methods [35, 40]. Here, we use in a first attempt the unsteady Reynolds averaged Navier Stokes equations (RANS) to compute the complete injection process and direct numerical simulations (DNS) to analyse the microscale processes, bubble growth, interaction, and spray breakup. For large eddy simulations (LES) or RANS two general approaches exist to model two phase flows. One possibility is to treat the continuous phase in an Eulerian framework and the dispersed phase as Lagrangian particles. However, this poses the problem that prior to spray breakup the continuous phase represents the liquid and the vapour bubbles are treated as Lagrangian particles, while after spray breakup the attribution of the two phases to the Eulerian and Lagrangian frameworks needs to be reversed. Alternatively, both phases are treated in an Eulerian framework. Then, the full set of mass, momentum, and energy equations are solved for each phase individually and coupled with each other through exchange terms. The two phases are distinguished by a mass or volume fraction with a transport equation that has to be solved additionally to the other conservation equations [4]. Assuming that the two phases are closely coupled and hence move at the same velocity allows to reduce the set of equations to one common momentum and mass transport equation, the so-called one-fluid approach. This simplification of the equation system requires the assumption of zero slip velocity between the two phases and that both phases experience a common pressure [7]. The one-fluid approach has already been successfully applied to simulate flashing sprays by several authors [19, 26, 30, 33, 34] and is therefore used for all large scale simulations of this work.

In RANS or LES, the nucleation, bubble growth, and formed droplets have to be modelled at the sub-grid scale. One commonly used model to describe flash evaporation on the sub-grid scale is the homogeneous relaxation model (HRM). This model considers the thermal non-equilibrium state of the flashing liquid by introducing a relaxation time and relating the evaporation rate to the deviation of the current mass fraction from an equilibrium value [3]. Finding a suitable description for the relaxation time is a challenging task, yet the model of Downar-Zapolski et al. [12], which is an empirical fit to flashing water experiments, has shown a general applicability to several fluids and injection conditions [18, 26, 29, 30]. Nevertheless, the model coefficients may require case specific adaptations which cannot be directly determined a priori and thus, model verification is required for each case [20, 33]. Approaches that attempt to model the effects at the phase interface, such as the Hertz-Knudsen model or momentum conservation like the Rayleigh-Plesset equation are also used to describe the bubble growth for flashing sprays [9]. Yet, these models only consider a singular bubble in an infinite liquid medium, neglecting the strong interaction of bubbles in flashing flows [10]. Further, these models require the knowledge of either the bubble number or the current mean diameter, which is typically unknown for large scale LES or RANS simulations. Due to these reasons there is a need for further investigation of the bubble growth and interaction of flashing sprays to improve the current models. One possible strategy to investigate the flashing process in detail, and to create a database is the use of DNS simulations.

Further modelling challenges arise due to shock structures that accompany the injection of highly superheated liquid into the combustion chamber. Shocks in flashing flows have been first associated with retrograde fluids [21, 39] and recently have also been observed in non-retrograde fluids, such as ethanol, acetone [23] or propane [31]. In the case of multi-hole injectors these shock structures can lead to severe changes of the spray morphology e.g. spray collapse [15, 19, 31] and therefore it is important to capture this phenomenon correctly. This highlights the complexity of simulating flashing processes which require knowledge of the thermodynamic processes on the smallest scales such as nucleation and bubble growth as well as capturing macroscopic effects as shocks which affect the spray shape.

The chapter is organized as follows: Sect. 2 introduces fully compressible DNS simulations that are used to deduce surface specific evaporation rates and to identify important simplifications with respect to domain size that bubble-bubble interactions induce. Section 3 provides DNS data on droplet size distributions after jet breakup and suggests a simple correlation for the interface generation during the flash-evaporation process. RANS simulations of the entire flash evaporation process including the injector and combustion chamber are then presented in Sect. 4. For all simulations, the computational setup mimicks conditions of the flash evaporation experiments conducted at the German Aerospace Center (DLR) Lampoldshausen [32]. There, liquid nitrogen with temperatures between 80 and 120 K is injected into a low pressure chamber with ambient pressures between 1000 and 1E+5.

2 DNS Simulation of Vapour Bubble Growth and Interaction

To investigate the behaviour of bubble growth and bubble-bubble interaction in detail, the vapour bubble growth is simulated with a DNS approach. A fully compressible two phase DNS code with a discontinuous Galerkin spectral element method is used [11] together with an approximative Riemann solver [14]. Further detail about the numerical framework can be found in Dietzel et al. [11] and Fechter et al. [14].

Accurate DNS simulations resolving all relevant scales, however, is extremely challenging and two important aspects are highlighted in the following: First, the DNS code has to account for compressibility effects to accurately capture the early stages of flash evaporation. As the vapour density and evaporation rate are directly linked to the local pressure it is important to include local pressure fluctuations induced by neighbouring evaporating bubbles. Secondly, direct simulation of the phase change term is unfeasible as it requires resolution on a molecular scale and thus the phase change terms appear in an unclosed form and require modelling. One possible model to describe the phase change at the interface is the Hertz-Knudsen model,

$$\begin{aligned} \dot{m}'' = \sqrt{\frac{1}{2\pi R}} \left( \lambda _e \frac{p_s(T)}{\sqrt{T_\textrm{L}}} - \lambda _c \frac{p_\textrm{G}}{\sqrt{T_\textrm{G}}}\right) , \end{aligned}$$
(1)

with the two coefficients \(\lambda _e\) and \(\lambda _c\) that are case specific. Therefore, these coefficients have to be determined for the case of flashing cryogenic liquid. This is done by simulating the growth of a single vapour bubble [11] and then applying the results to bubble arrays [10].

2.1 Single Vapour Bubble Growth

To calibrate the Hertz-Knudsen model (HKM) such that it can be used in a full 3D DNS simulation with multiple bubbles, the growth of a single bubble is compared first with the solution of the coupled Rayleigh-Plesset and energy equations [25]. A direct calibration of the two parameters \(\lambda _c\) and \(\lambda _e\) in the HKM is difficult and it is therefore conventional to set \(\lambda _c = 0.99\lambda _e\). The slight reduction of the condensation coefficient is introduced to stabilize the simulation at the early stages of bubble growth while the effect on the entire time interval is negligible. Further, as the early bubble growth stage has only a negligible contribution to the overall volume change, and as the bubble growth rate can vary over several magnitudes the parameter \(R_0^*\) is introduced to speed up the simulations. This parameter relates the starting radius of the bubble, \(R_0\), in the 3D simulation to the critical radius \(R_\text {crit}\) with,

$$\begin{aligned} R_0^* = \frac{R_0}{R_\text {crit}}. \end{aligned}$$
(2)

The critical radius equals the smallest radius for a stable nuclei and can be calculated from [5],

$$\begin{aligned} R_\text {crit} = \frac{2\sigma }{p_\textrm{sat}(T_\infty )-p_\textrm{L}}. \end{aligned}$$
(3)

This gives the three independent simulation parameters: the pressure p, liquid temperature \(T_\textrm{L}\) and starting radius \(R_0^*\). It is more conventional to characterize flashing flows by a superheat ratio \(R_p = p_\text {sat}/p\) (substituting p) and \(\lambda _e\) is then calibrated as a function of superheat ratio, liquid temperature and starting radius.

The results show that the HKM can be calibrated to match the Rayleigh-Plesset solution within 10% of the integrated mass flux. The difference in mass flux, however, is also due to minor density changes between the two models. Even though the Hertz-Knudsen model can be calibrated for each computed single bubble case, the evaporation coefficient is case dependent and a simple functional expression with the input parameters, \(R_p\), T, and \(R^*_0\) seems elusive [11]. This is due to the high non-linearity of the growth rate over the bubbles growth history and the changing driving forces for bubble growth. Early growth is dominated by inertia effects while later stages are dominated by heat diffusion. However, as indicated above, the later stages of the growth process determine the volumetric expansion (and thus spray breakup) and a representative \(\lambda _e\) can be approximated by

$$\begin{aligned} \lambda _e = \lambda _{e,\text {RPE}} 1.27 (1-\exp (-0.0086R_{end}^*))^{0.61}&\textrm{with} &\lambda _{e,\text {RPE}} = \frac{\dot{m}''(R_{end})\sqrt{2\pi R}}{\frac{p_\text {sat}}{\sqrt{T_\textrm{L}}}-0.99\frac{p_v}{\sqrt{T_v}}} \end{aligned}$$
(4)

and \(R^*_{end} = 10R^*_0\) [11]. Note that \(R^*_{end}\) denotes the bubble radius when the bubbles start to merge and is related to the nucleation rates. This will be further discussed in Sect. 3.

2.2 Bubble Interaction

In the previous section we have introduced a suitable calibration for the Hertz-Knudsen model that can now be applied to full 3D DNS simulations of multiple growing bubbles within a superheated jet. The initial conditions of the setup mirror the relevant conditions of the flash boiling experiments conducted at the German Aerospace Center (DLR) in Lampoldshausen using cryogenic liquids injected into chambers at near vacuum conditions. Resolving the injector would require at least 2000 cells for the radius alone, thus rendering a full scale simulation unfeasible. Consequently, the bubble growth and interaction is studied with a simplified setup that considers only the tip of the jet. A comparison of 1D bubble columns, 2D liquid jet slices, and 3D simulation of the jet tip showed that the simplified 2D simulation captures the bubble dynamics, interactions and jet expansion sufficiently accurate [10]. For all simulations the bubble spacing is set to 10 times the starting radius \(R^*_0\) as the last decade of bubble growth determines 99.9% of the volumetric expansion. Thus, \(R^*_0\) determines the bubble size when bubbles coalesce and the jet breaks up and it is directly related to the bubble density (or nucleation rate). We note here that no expression exists that would allow for an accurate estimate of the nucleation rate as known expressions can differ by more than 5 orders of magnitude. It is thus justified to impose a nucleation rate (or \(R^*_0\) or \(R^*_{end}\)) in the DNS and use it as the only free parameter in RANS or LES to match full scale simulations with experiments.

For illustration, the simulation results are shown in Fig. 1 for a 3D reference case utilizing two symmetry planes and one dimension of freedom with the initial temperature of 120 K, superheat \(R_p=5\) and radius \(R^*_0 = 50\). The liquid is located in the lower two-thirds of the column, the jet’s interface is indicated by the pale blue rectangle and symmetry boundary conditions are used at the bottom of the domain. It is immediately apparent that the outer most bubbles grow magnitudes faster than the inner bubbles. This is due to pressure variations in the bubble column. The bubble expansion induces pressure waves yielding much higher pressures when moving away from the jet’s interface and thus reducing evaporation rates and bubble growth for the inner bubbles. This finding is quantified in Fig. 2 with a series of 3D simulation with up to 125 bubbles. The figure shows normalized bubble growth as function of their dimensionless distance to the jet surface \(x/D_\textrm{bub}\). The normalization factor is the bubble growth of a single bubble. We note that even the expansion of the first bubble, as seen from the jet surface, is reduced by more than 20% and that radial growth of a bubble in the “fifth row” is reduced by 90 %. The importance of this finding cannot be sufficiently stressed: bubbles that are further away than \(x/D_\textrm{bub} \approx 5 - 10\) from the jet surface at time of jet breakup have hardly grown and do not need to be considered for the jet breakup dynamics [10].

Fig. 1
figure 1

Adapted from [10] with permission from Elsevier

Contour plots of the bubble column test case for three different time steps with five bubble rows along the x-direction. The left and right planes show the time averaged pressure and the density fields, respectively. The blue coloured iso-surfaces indicate the interfaces between bubbles and superheated liquid.

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Fig. 2
figure 2

Adapted from [10] with permission from Elsevier

Bubble expansion in 3D compared to the single bubble growth \((R_\text {sb})\) in respect to their normalized distance to the jet surface.

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3 Spray Break-up

The simulations in Sect. 2 demonstrate that analysis of small bubble configurations is sufficient to analyse breakup and subsequent droplet formation. The compressible DNS cannot, however, capture the final stages of breakup and droplet formation and additional DNS are needed. These final stages are likely to be within the inertia driven stage [28] and are dominated by the dynamics at the time of the first bubble coalescence when liquid structures burst and the first droplets form. It is therefore justified to calibrate the fluid properties as well as the bubble growth rate towards this instance in time and use an incompressible, multiphase solver such as Free-Surface 3D - FS3D [13] as only the efficiency of the incompressible treatment allows for the entire breakup dynamics and sufficient resolution of satellite droplets and droplet distributions. It follows that—similar to the DNS presented in Sect. 2—the ambient pressure p, liquid temperature T, and final bubble merging radius \(R_{end}\) (or nucleation rate) determine the numerical setup. Details on the setup and also limitations of the setup are discussed in Loureiro et al. [28].

3.1 Test Case Setup

The test cases examine a liquid blob that either represents an outer layer of the liquid jet or has detached from the jet at the injector exit where a sudden pressure drop occurs. In that location high superheat ratios are reached which lead to high nucleation rates and bubble growth. A first study [28] where a regular cubic lattice distribution of the vapour bubbles was investigated, identified the three distinct breakup regimes that were labelled “retracting liquid” (\(\text {We}\le 2\)), “ligament stretching” (\(2<\text {We}\le 20\)), and “thin lamella” (\(\text {We}\ge 20\)) regimes. These regimes are categorized by the characteristic Weber and Ohnesorge numbers,

$$\begin{aligned} \text {We}&= \frac{2R_\text {end}\rho _\textrm{L} \dot{R}^2}{\sigma }&\textrm{and}&\textrm{Oh}&= \frac{\mu }{\sqrt{2R_\text {end}\rho _\textrm{L}\sigma }}. \end{aligned}$$

Here, \(\sigma \) is the surface tension and \(\dot{R}\) the bubble growth rate determined by the Rayleigh-Plesset solution as a function of \(\dot{R}=f(p,T,R_{end})\).

The regular cubic lattice arrays provide a unique characterization of droplet distance, they lead however, to a typically bi-disperse droplet size distribution with the larger droplet size being imposed by the interstitial volume and the smaller one by ligament shedding and local surface tension. Here, we therefore distribute the bubbles randomly to prevent any systematic bias for the droplet size. An example of the spray breakup process in the ligament stretching regime with 512 initial bubbles is shown in Fig. 3.

3.2 Droplet Size Distribution

The process of bubble coalescence, pinching, ligament stretching and retraction to form satellite droplets is qualitatively similar to the process observed for the cubic lattice array [28] and thus confirms the regime classification. A typical droplet distribution is shown in Fig. 4. Across the parameter range investigated here, we note that the mean diameter is still close to the value that would be imposed by the interstitial volume but subsequent pinching, stretching and retraction leads to a more realistic, much wider distribution that can be approximated by a Gaussian. Note that the area weighted (Sauter) diameters are shown. This reduces the effect of the very small droplets that are generated when stretched ligaments pinch. This droplet size will be determined by the cell size and is likely to be a numerical artefact. However, their mass is negligible and they will contribute very little to any subsequent combustion process. The figure further includes an analytical approximation for the droplet diameter, \(D_\textrm{ref}\), that is based on the relation of the kinetic energy in the liquid to the droplets volumetric surface energy. It is given by

$$\begin{aligned} D_\textrm{ref} = \frac{8\sigma }{\rho _\textrm{L}\dot{R}^2} = \frac{16}{\text {We}}R_\textrm{end}. \end{aligned}$$
(5)

A comparison with the computed Sauter mean diameter allows to derive an exponential fit based on a linear least-square regression of \(\log (\overline{D}_A/D_\textrm{ref})\),

$$\begin{aligned} \frac{\overline{D}_A}{D_\textrm{ref}} \approx 0.294 \sqrt{\text {We}}. \end{aligned}$$
(6)

This expression provides an estimate of the droplet size which is only based on the geometric setup and the fluid properties.

Fig. 3
figure 3

Time sequence of the spray breakup process and droplet generation. Break-up in the ligament stretching regime with conditions \(T_\textrm{L} = {120} K\), \(R^*_f=5\), \(\text {We}=3.62\), Oh \(=0.104\)

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Fig. 4
figure 4

(\(T_\textrm{L} = {120} K\), \(R^*_f=16\), \(\text {We}=13.38\), Oh\(=0.05\)): Area weighted droplet size distribution. The limit of the mesh resolution of 4 cells for a droplet is marked by the line \(4\Delta x\)

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Further analysis of the DNS (not shown here) provides an evolution for the surface area of the interface between liquid and vapour phase. Loureiro et al. [27] observed that (i) surface density is largely determined by bubble density (i.e. nucleation rate) and Weber number and (ii) stays relatively constant after the majority of bubbles have started to coalesce as surface creation by ligament pinching and surface destruction by ligament contraction seem to balance. A functional dependence for the specific surface area as function of time and We could be derived.

4 RANS Simulation of Flashing Sprays

The simulation of the entire injection process of flashing cryogenic liquids with resolved interfaces is still unfeasible due to the wide range of scales involved. Therefore, RANS or LES are a suitable alternative to get engineering relevant and yet accurate information, but bubble growth and bubble/droplet dynamics occur at sub-grid scale and need to be modelled. Models can be based on the findings highlighted in Sects. 2 and 3 or on standard closures that use empirical fitting constants. As outlined in the introduction, the present simulations of the complete injection process following the fluid from a high-pressure reservoir, through the injector, to the combustion chamber are based on an Euler-Euler approach. To simulate the cryogenic flash boiling a new solver has been developed in OpenFOAM [17, 36].

4.1 Governing Equations

The governing equations for volume fraction, momentum and energy transport are solved. In contrast to pure one-fluid solvers the energy equation is solved for each phase separately, hence allowing a temperature difference between the two phases at a single location. The choice to model the energy equation for both phases separately is motivated by the assumption, that the vapour is generated at the saturation conditions of the local pressure, thus the temperature of the liquid is much higher than that of the vapour. As the evaporation rate model depends on the liquid temperature alone, representation of the correct temperature difference is important [17]. Further, real gas effects that appear in the modelling of the heat flux are not accounted for as the very low pressure of the vapour justifies ideal gas assumptions. All other fluid properties, such as saturation conditions, liquid density or viscosity are determined using the thermodynamic library CoolProp [2]. To reduce the computational time the thermodynamic properties are calculated in advance and tabulated, such that required properties can be looked up during the run time of the simulation. More detailed information about the derivation of the governing equations can be found in Gärtner et al. [17].

4.1.1 Compressibility Modelling

Modelling the compressibility for the simulation of flashing flows is important, as higher superheated sprays exhibit behaviours similar to under-expanded gaseous jets with the same shock patterns [22, 31]. Typically, supersonic jets are simulated by solving for the density field and obtaining the pressure with an appropriate equation of state. Yet, this kind of solvers is not well suited for all Mach number conditions, as would be present for the simulation of flashing sprays. Therefore, a pressure based approach is chosen [8]. The pressure equation, as combination of momentum and mass conservation, can then be written as

$$\begin{aligned} \frac{\partial \psi p}{\partial t} + \nabla \boldsymbol{\cdot }\left( \psi p \frac{\textrm{H}(U)}{a_p}\right) - \nabla \boldsymbol{\cdot }\left( \frac{\rho }{a_p}\nabla p\right) = 0, \end{aligned}$$
(7)

where \(\psi \) is the compressibility of the fluid, \(a_p\) are the coefficients of the velocity matrix and \(\textrm{H}(U)\) represents the discretized momentum equation except for the pressure gradient. For supercritical flows this is a suitable way as vapour and liquid densities can be expressed with \(\rho = \psi p\) [37], but it is no longer suitable for cryogenic liquids. Further, the solution of this equation gives a mass flux whereas the transport of the volume fraction requires a volume flux. Therefore, with the goal of preserving the volume change in flashing flows which is responsible for the large spray angles, the pressure equation of the developed solver is derived by first extracting the density from the convective term of the continuity equation giving,

$$\begin{aligned} \frac{\textrm{D}\rho }{\textrm{D}t} + \rho \nabla \boldsymbol{\cdot } (\boldsymbol{u})&= 0,\end{aligned}$$
(8)
$$\begin{aligned} \frac{\textrm{D}\rho }{\textrm{D}t} +\rho \nabla \boldsymbol{\cdot }\left( \frac{\textrm{H}(U)}{a_p}\right) - \rho \nabla \boldsymbol{\cdot }\left( \frac{\nabla p}{a_p}\right)&= 0. \end{aligned}$$
(9)

The material derivative of density can be replaced by the total derivative of density as a function of mass fraction, \(\chi \), pressure p, and enthalpy h,

$$\begin{aligned} \frac{\textrm{D}\rho }{\textrm{D}t}=\underbrace{\left( \frac{\partial \rho }{\partial \chi }\right) _{p,h} \frac{\textrm{D}\chi }{\textrm{D}t}}_{\textbf {evaporation}} + \underbrace{\left( \frac{\partial \rho }{\partial p}\right) _{\chi ,h} \frac{\textrm{D}p}{\textrm{D}t}}_{\textbf {compressibility}} + \underbrace{\left( \frac{\partial \rho }{\partial h}\right) _{\chi ,p} \frac{\textrm{D}h}{\textrm{D}t}}_{\textbf {heat expansion}} . \end{aligned}$$
(10)

The compressibility of the vapour-liquid mixture is much higher than that given by the pure phase values and can be expressed by an inversely density weighted average [4]. In analogy, the change of density with respect to enthalpy can be described and finally combining the changes due to pressure and enthalpy gives

$$\begin{aligned} \frac{\textrm{D}\rho }{\textrm{D}t} = \left( \frac{\partial \rho }{\partial \chi }\right) _{p,h} \frac{\textrm{D}\chi }{\textrm{D}t} + \rho \left( \frac{\alpha _\textrm{L}}{\rho _\textrm{L}}\frac{\textrm{D}\rho _\textrm{L}}{\textrm{D}t} +\frac{\alpha _\textrm{G}}{\alpha _\textrm{G}}\frac{\textrm{D}\psi _\textrm{G} p}{\textrm{D}t} \right) . \end{aligned}$$
(11)

Here, \(\alpha _\textrm{L}\) denotes the liquid volume fraction, \(\alpha _\textrm{L} = V_\textrm{L}/V\). Note that the density of the liquid is not replaced to avoid any numerical instabilities caused by the very low compressibility of the liquid. This leaves the first term of Eq. (11) to be modelled.

The turbulent contributions arising from the Reynolds averaging of the governing equations are modelled with a k-\(\omega \) SST model for the momentum transport and with the Reynolds analogy for the turbulent diffusion of the scalar and energy transport equations. Further details can be found in Gärtner et al. [17].

4.1.2 Phase Change Models

Several models to predict the phase change in flashing flows exist. We first introduce the homogeneous relaxation model (HRM) as the standard model that is used to compute a reference case and to analyse the general structure of the flashing spray. The HRM model relates the rate of phase change to the deviation of the mass fraction from the local equilibrium value,

$$\begin{aligned} \frac{\textrm{D}\chi }{\textrm{D}t} = - \chi \frac{h_\textrm{L}(p,T) - h_\textrm{SL}(p)}{h_\textrm{SG}(p) - h_\textrm{SL}(p)} \frac{1}{\Theta } = \frac{\dot{m}_\textrm{L}}{\rho }, \end{aligned}$$
(12)

with h denoting the enthalpies of liquid, the saturation state and the latent heat (depending on its subscript). The model was originally proposed by Bilicki and Kestin [3] and a formulation for the relaxation time, \(\Theta \), was found by Downar-Zapolski et al. [12]. In their work the authors derived a high and low pressure fit of the relaxation time based on flashing water experiments,

$$\begin{aligned}&\text {low pressure fit:}&\Theta&= \Theta _0 \epsilon ^\beta \left( \frac{p_\text {sat}(T_\textrm{L}) - p}{p_\text {sat}}\right) ^\lambda ,\end{aligned}$$
(13)
$$\begin{aligned}&\text {high pressure fit:}&\Theta&= \Theta _0 \epsilon ^\beta \left( \frac{p_\text {sat}(T_\textrm{L}) - p}{p_c - p_\text {sat}}\right) ^\lambda . \end{aligned}$$
(14)

Here, \(\epsilon \) is the void fraction, \(p_\text {sat}(T_\textrm{L})\) the saturation pressure based on the liquid temperature, \(p_c\) the pressure at the critical point and \(\Theta _0\), \(\beta \), and \(\lambda \) are empirical coefficients. Despite being fitted to flashing water experiments, the description of the relaxation time derived by Downar-Zapolski et al. [12] have shown a wide range of applicability for several technical applications, fluids, and thermodynamic conditions. Because of these reasons, the HRM model is applied for all LES/RANS simulations to model the evaporation rate.

An alternative model could be derived from the DNS in Sects. 2 and 3. The calibrated Hertz-Knudsen model provides an expression for the surface specific evaporation rate of the bubble. The phase interface is, however, not resolved in RANS or LES and requires modelling itself. Transport equations for the volume specific interface surface exist, they are commonly referred to as Euler-Lagrange spray atomisation models (ELSA) [1, 24] and are available in OpenFOAM. Existing closure for the transport equation of the sub-grid surface area employ source terms for interface generation that account for turbulent liquid jet breakup due to viscous effects at the interface, but they do not include surface generation through bubble growth and coalescence that govern the dynamics of flash atomization processes. Here, the correlations developed in Sect. 3 could be used. In addition, droplet size distribution would need to be specified after local disintegration of the jet and could also be deduced from the DNS introduced in the same section. However, a reference solution is sought first and results from the HRM model will be shown in Sect. 4.3.

4.2 Numerical Methods

The system of equations is solved by a combined semi-implicit method for pressure linked equations (SIMPLE) and a pressure implicit with splitting operator (PISO) loop called PIMPLE. The discretization schemes are run time selectable. Typically, total variation diminishing (TVD) schemes such as van Leer limiter or OpenFOAM’s limited linear are chosen for problems involving two phase flows. However, despite having formal second order accuracy they cannot achieve this accuracy in real applications [6, 38]. Further, the implemented schemes in OpenFOAM do not guarantee proper resolution of the shock front for all cases [17]. To solve problems with discontinuities while maintaining a high order of accuracy, weighted essential non-oscillating (WENO) schemes are a suitable choice. To include this scheme in OpenFOAM a computationally improved version was developed in Gärtner et al. [16] and is used here for the momentum, enthalpy, and kinetic energy transport. Lastly, the volume fraction transport is solved with a multi-dimensional limited with explicit solution (MULES) procedure, which is an effective method to guarantee the boundedness of the volume fraction. As the current work focuses on quasi stationary states, a first order in time Euler integration is applied.

4.3 Shock Structure

The solver’s capability to resolve low and high Mach number regimes in flashing flows has been validated by comparison with experiments of flashing acetone sprays where the shock structures were visualized and where unambiguous measurements of the shocks’ positions exist [23]. Gärtner et al. [17] demonstrated that the computed lateral expansion of the shocks is accurate and that the solver is capable to predict the shock size and position well.

After validating the solvers capability to capture supersonic conditions accurately, the two cryogenic nitrogen sprays presented in Table 1 are simulated and compared to shadowgraph images provided by DLR Lampoldshausen [17]. The simulation domain consists of a two dimensional wedge of the injector geometry and the attached vacuum chamber with a minimum cell size of 10 \(\upmu \)m.

A comparison of the velocity magnitude and the density gradient between simulation and the shadowgraph images is presented in Fig. 5. Here, the characteristic bow shock of an under-expanded jet is visible. In addition, a region with low velocity is observed about 20 to 30 jet diameters downstream of the injector. This region of low velocity is caused by the relative high velocity of the slip stream and the low velocity in the core of the spray, which causes a shear layer that leads to a recirculation zone in that region. Even though no velocity data is available from the experiments, the shadowgraph images indicate the presence of such a recirculation zone by nearly motionless or slightly upstream floating dark structures. This is visualized in more detail for case LN2-2 in Fig. 6, where the velocity direction is marked by vectors and the recirculation zone of the experiment is marked by a yellow square. Within this recirculation zone droplets can accumulate and cause the dark structures visible in the shadowgraph images. In conclusion, the recirculation zone visible in the numerical results gives an explanation for the observed structures in the shadowgraph images and reinforces the importance to include the compressible effects.

Fig. 5
figure 5

Adapted from Gärtner et al. [17] with permission from Elsevier

Velocity magnitude and magnitude of the vapour density gradient for the flashing liquid nitrogen cases compared to the shadowgraph images. The location of the nearly motionless structures are marked with a yellow square.

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Fig. 6
figure 6

Adapted from Gärtner et al. [17] with permission from Elsevier

Shadowgraph image of case LN2-2 compared to the mass fraction field and the velocity vectors coloured in respect to their velocity magnitude. The recirculation zone is marked with a yellow box in the shadowgraph and simulation image.

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Additional to the shadowgraph images, the mass flow rate is provided and can be used to validate the evaporation rate inside the injector. A comparison of the mass flow rate of the simulation and experiment show that the mass flow rate predicted by the simulation is within the experimental uncertainty, see Table 1. Further, the results confirm the simulation data that the liquid nitrogen already evaporates within the injector, thus causing the pressure to increase and reducing the mass flow compared to a non-flashing flow. Hence, including the injector and the upstream conditions is important to capture the spray conditions correctly.

5 Conclusions

Flash evaporation of cryogenic liquids is investigated numerically using DNS and RANS approaches. DNS simulations of the vapour bubble growth, interaction and succeeding spray breakup were performed to gain a detailed insight into the bubble and droplet dynamics. In addition, RANS simulations featuring the full injection process of the flow through the injector and in the combustion chamber have been realized. Details on the numerics have been published in previous work  [10, 11, 15, 17, 27, 28].

Table 1 Boundary conditions of flashing cryogenic nitrogen sprays
Full size table

The key results are:

  • The Hertz-Knudsen model can predict the volume change of a growing bubble in 3D simulations.

  • Only bubbles close to the jet surface grow significantly as the bubble expansion causes pressure increases and thus reduced superheat and growth rates in the interior of the jet. Bubbles at distances that are more than 10 bubble diameters away from the jet interface can be neglected as they do not contribute to the jet expansion and breakup.

  • The characteristics of the spray breakup depend on the Weber and Ohnesorge number and fall in regimes that can be labelled retracting liquid, ligament stretching, and thin lamella regime.

  • The order of magnitude of the resulting mean droplet diameter can be estimated by a simple kinematic relationship.

  • The mean Sauter diameter is largely determined by the interstitial space between the bubbles at time of coalescence and the droplet size distribution is close to Gaussian.

  • Surface area generation by bubble growth and spray breakup is largely determined by bubble number density and We. Secondary breakup leads to slow changes in surface area only.

  • A RANS based solver has been developed and validated. Shock structures can be well captured.

  • Simulations and experiments agree with respect to the major flow features.

Future work will provide more quantitative comparison (e.g. droplet velocities and sizes) between simulations and experiments. The simple HRM model may not allow to extract such information. An LES code with a one-fluid two-equation model for the two-phases (one for the volume fraction and one for the surface area) exists. The DNS results will need to be exploited to develop and implement a suitable source term for the surface area transport equation that accounts for the specific characteristics of flash atomization. Dietzel et al. [10] provide the surface area specific growth rates as functions of local (LES-filtered) pressures and the DNS by Loureiro et al. [27] provide surface generation terms due to the dynamics of the breakup process.