A Subgrid-Scale Model for Large-Eddy Simulation of Liquid/Gas Interfaces Based on One-Dimensional Turbulence

Abstract

The interface/turbulence interaction between two fluids in a turbulent environment has an important role in many technical processes, e.g. primary liquid atomization in combustion devices. Primary atomization has a significant role in spray formation and its characteristics. The resulting dynamics typically span 4–6 orders of magnitude in length scales, making detailed numerical simulations exceedingly expensive. This motivates the need for modeling approaches based on spatial filtering such as large-eddy simulation (LES). In this paper, a new approach based on One-Dimensional turbulence (ODT) is presented to describe the subgrid interface dynamics. ODT is a stochastic model simulating turbulent flow evolution along a notional one-dimensional line of sight by applying instantaneous maps that represent the effects of individual turbulent eddies on property fields. It provides affordable high resolution of interface creation and property gradients within each phase, which are key for capturing the local behavior as well as overall trends. ODT has previously been shown to reproduce the main features of an experimentally determined regime diagram for primary jet breakup. Here a new approach called VODT is presented which produces a size-conditioned as well as a total time rate of generation of droplets for given flow conditions at an interface. At the LES level, the total droplet generation from VODT is interpreted as a rate of mass conversion of LES-resolved liquid into unresolved droplets. Preliminary results of applying VODT to a cell with a planar-shear-layer are discussed at the end of the paper.

1 Introduction

The interaction between turbulence and interfaces of immiscible fluids is seen in many engineering applications, e.g., primary atomization in combustion devices. The atomization process plays a significant role in combustion performance, including efficiency and emissions. Despite its importance, a detailed description of primary atomization has remained elusive, due in part to insufficient understanding of how interfaces modulate the surrounding flow field and undergo breakup. Many attempts have been made to describe interfacial instabilities using linear stability analysis, however, these only provide a very limited picture of interfacial dynamics, especially in turbulence and complex geometries. Because of this, there remains a need to understand and model the interactions between two immiscible fluids in a turbulent environment.

Predictive simulations with high spatial and temporal resolution, i.e., Direct Numerical Simulations (DNS), offer an alternative way to study liquid-gas interface dynamics during primary breakup. But despite the significant benefits provided by DNS, the large computational cost precludes their use in many flows of engineering interest. Therefore, there is a need for appropriate interface dynamics models lower the computational cost of predicting the atomization process. While requiring physical models for the small unresolved scales of the flow, Large-Eddy Simulation (LES) has shown to be a useful tool that can provide much more flexibility on resolution coarser than DNS by introducing a spatial filter into the governing equations and resolving only the scales larger than the filter width [3]. However, the LES sub-filter models typically neglect the contribution of the surface tension term and are based on a cascade process hypothesis that may be questionable in the context of surface tension-driven atomization. This leads the need for a new LES subgrid interface dynamics model.

A One-Dimensional Turbulence (ODT) model is considered here as an affordable model for simulating large Reynolds and Weber number flow configurations. ODT is a stochastic model simulating turbulent flow evolution along a notional one-dimensional line of sight by applying instantaneous maps that represent the effects of individual turbulent eddies on the flow properties. ODT has recently been used by the authors [5, 6] to reproduce the main features of an experimentally determined regime diagram for primary jet breakup.

ODT can be used both as a stand alone tool and as a sub-grid model for LES or RANS. This creates a possibility to use ODT as a subgrid resolution model in LES simulations to describe/model subgrid interface dynamics. This approach is described in detail in the following sections.

2 Governing Equations

The flows investigated in this study are governed by the incompressible Navier-Stokes equations for immiscible two-phase flow. The momentum equation is given by

$$\begin{aligned} {\dfrac{\partial \rho u}{\partial t} + u \cdot \nabla \rho u = -\nabla p + \nabla \cdot [\mu (\nabla u + \nabla ^{T} u)] + \sigma \gamma \delta (x - x_{\varGamma })n }, \end{aligned}$$

(1)

where u is the velocity, $\rho$ the density, p the pressure, and $\mu$ is the dynamic viscosity. The last term in Eq. 1 is the singular surface tension force where $\gamma$ denotes the curvature of the interface, $\delta$ is the Dirac delta function, $x_{\varGamma }$ is the point on the interface $\varGamma$ closest to the point x and n is the interface normal vector.

To compute the phase interface, in addition, a transport equation for the liquid volume fraction $\alpha$ in a computational cell is solved

$$\begin{aligned} {\dfrac{\partial \alpha }{\partial t} + \nabla \cdot ( u \alpha ) = 0}. \end{aligned}$$

(2)

The interface unit vector n and the interface curvature $\gamma$ can be theoretically expressed in terms of the liquid volume fraction as

$$\begin{aligned} {n = \dfrac{\nabla \alpha }{|\nabla \alpha |}, \quad \gamma = \nabla \cdot n }. \end{aligned}$$

(3)

Following the continuum surface force approach [2] the surface tension force in Eq. 1 is modelled as

$$\begin{aligned} {\sigma \gamma \delta (x - x_{\varGamma })n = \sigma \gamma \nabla \alpha }. \end{aligned}$$

(4)

LES formulation is based on applying spatial filtering to Eq. 1. The filtered structures or large scales of the flow field are resolved directly, though the terms filtered out need to be modeled. These terms are called sub grid scale (SGS) terms. If they are small enough they behave isotropic and geometry independent. The SGS terms can be closed by applying a model e.g. the Smagorinsky model. However, these models usually neglect the effect of the sub-filter surface tension term and need to be reconsidered.

In particular, LES does not see interface wrinkles below its resolution scale, so the LES-resolved interface is much smoother and has less total surface area than the true interface. Therefore it doesn’t fully account for the true total amount of stored surface-tension area. As Fig. 1 illustrates, when the Weber number is high enough there are some scales of interface wrinkling that are not resolved by the LES and cause droplet generation. The following sections describe how ODT can be used to model these unresolved scales.

Fig. 1

Using ODT as a sub-filter model for LES

3 One-Dimensional Turbulence (ODT)

The ODT model of Kerstein used in this study is briefly described in this section. For a fully detailed description we refer to Kerstein et al. [4], Ashurst et al. [1], and its extension to modelling primary breakup by Movaghar et al. [6].

ODT is a stochastic model simulating the evolution of turbulent flow along a notional line of sight through a three-dimensional flow. Here ODT line is oriented in the transverse y direction or normal to the interface. This setup provides high lateral resolution of the relevant physics near the interface. On the other hand, one-dimensional formulation provides the affordability needed to capture the full range of scale separation at high Reynolds and Weber numbers.

In ODT formulation the fields on the 1-D line of sight are evolved by two different mechanisms: molecular diffusion or time advancement and eddy events representing advection.

Time advancement of the present ODT formulation is expressed schematically as

$$\begin{aligned} {\dfrac{\partial u_{i}}{\partial t} - {\nu } \dfrac{\partial ^{2} u_{i}}{\partial y^2} = Eddies}, \end{aligned}$$

(5)

where $\nu$ is the kinematic viscosity and the indices $i = 1$, 2, 3 denote the streamwise, lateral and spanwise velocity components, respectively, corresponding to the spatial coordinates (x, y, z). This equation formally represents the two processes that can change the value of $u_{i}$ at a given location y and time t.

The eddy events representing advection may be interpreted as the model analog of individual turbulent eddies. In ODT, an eddy of size l is represented by an instantaneous map acting on each property field within an interval $[y_{0}, y_{0}+l]$ on the line. The mathematical formulation of the map should satisfy measure preservation and continuity of mapped profiles. The new velocity field after the map event is given by,

$$\begin{aligned} \hat{u}_{i}(y,t) = u_{i}(f(y),t) \end{aligned}$$

(6)

where the inverse of the triplet map is specified by

$$\begin{aligned} \tiny \small { f(y) = y_{0} + {\left\{ \begin{array}{ll} 3(y-y_{0}), &{} \text {if } y_{0}\le y \le y_{0}+(1/3)l, \\ 2l-3(y-y_{0}),&{}\text {if } y_{0}+(1/3)l\le y \le y_{0}+(2/3)l, \\ 3(y-y_{0}) - 2l, &{} \text {if } y_{0}+(2/3)l\le y \le y_{0}+l, \\ y-y_{0}, &{} \text {otherwise } \end{array}\right. }} \end{aligned}$$

(7)

If the eddy range $[y_{0}, y_{0}+l]$ contains no phase interfaces, then it is a single-phase eddy whose implementation is the same as in previous ODT formulations. If instead the eddy range contains an interface then it is a multiphase eddy requiring the domain-integrated energy conservation. This requires that the change of kinetic energy after the eddy event, $\varDelta E_{kin}$ is equal and opposite to any surface-tension potential-energy change, $\varDelta E_{\sigma }$ caused by triplet mapping of phase interfaces.

For this purpose, the kernel functions $c_{i}K(y)$ and $b_{i}J(y)$ are added to the ${u_{i}}$ profile created formerly by triplet mapping. On this basis Eq. 6 is rewritten as

$$\begin{aligned} u_i(y) \rightarrow u_i(f(y)) + b_iJ(y)+c_iK(y) \end{aligned}$$

(8)

and

$$\begin{aligned} \rho (y) \rightarrow \rho (f(y)), \end{aligned}$$

(9)

where f(y) is the inverse of the triplet map, $K = y- f(y)$, $J=|K|$, and $b_i$ and $c_i$ are assigned based on physical modeling. The requirement $\int K(y)dy = 0$ enforces momentum conservation.

ODT samples eddy events from an instantaneous distribution that evolves with the flow. These events are individually parameterized by position $y_{0}$ and size l.

The number of events during a time increment dt for eddies located $[y_{0},y_{0}+dy]$ in the size range $[l,l+dl]$ is denoted $\lambda (y_{0},l;t)dy_{0}dldt$, where the event rate density $\lambda$ is defined as

$$\begin{aligned} \lambda (y_{0},l;t) = C/(l^{2}\tau (y_{0},l;t)). \end{aligned}$$

(10)

with dimensions of $events /(location\times size \times time)$. The adjustable parameter C scales the overall eddy frequency and $\tau$ is the eddy time scale. To find this eddy time scale, the square of the velocity implied by l and $\tau$ is modeled as

$$\begin{aligned} (l/\tau )^{2} \sim E_{final} - Z(\nu ^{2}/l^{2}). \end{aligned}$$

(11)

On the right hand side, the first term is the final value of the available kinetic energy, denoted $E_{kin}$ in the absence of surface-tension effects, $E_{final}$ = $E_{kin} -\varDelta E_{\sigma }$ and the second term involving the parameter Z suppresses unphysically small eddies, such as those smaller than the Kolmogorov scale. Following [6], $\varDelta E_{\sigma }$ is formulated as

$$\begin{aligned} \varDelta E_{\sigma } = - { 4 \sigma \over \rho _\mathrm{eddy} l}. \end{aligned}$$

(12)

Because $\varDelta E_{\sigma }$ and the last term in Eq. 11 are both negative, the right-hand side of that equation can be negative but the left-hand side must be positive. This is an indication that the selected eddy is energetically forbidden, corresponding to $\lambda = 0$ for such eddies.

4 Virtual ODT for LES/VOF Closure

As described, ODT can be used as a stand alone computation tool. This creates a possibility to use ODT to compute a subgrid breakup table. As shown in Fig. 1 it is assumed that the ODT closure of LES/VOF can be formulated on an ODT domain locally normal to a given interface element in every interface LES cell. For generality the interface is allowed here to be at any location $y=I$ on the ODT domain where I is the interface location. ODT produces a size-conditioned as well as a total time rate of generation of droplets at a given VOF interface. At the LES level, the total droplet generation is interpreted as a rate $\dot{M}$ of mass conversion of LES-resolved liquid into unresolved droplets that are then deemed to reside in the gas phase. Accordingly, for an interface element within a LES cell, droplet generation causes recession of the interface at a speed $\dot{M} / (A \rho _l)$ inserted as a source term into Eq. 2 and updates the LES governing equations. A reduced formulation of ODT, called virtual ODT (VODT), that is suitable for economical closure in this manner is introduced next.

As discussed earlier, ODT contains two main mechanisms, viscous time advancement and eddy events. For the tabulation purposes, we suppose there is no viscous advancement but only eddy sampling. Nevertheless the accepted eddies are not implemented and just their statistics are collected. Because the VODT state is not time advanced, the only effect of eddy events is droplet generation and the implied recession of the liquid surface. Therefore, eddies entirely contained in one phase have no effect, leaving only multiphase (hence droplet-forming) eddies to be considered. By not implementing the eddies, the eddy rate distribution is stationary. Thus the initial rate distribution is used to evaluate the rate of production of a droplet of any size. The effect of surface tension (and hence the We dependence), which does not appear in Eq. 5, is brought into the formulation through the physical modeling that specifies the eddy rate distribution.

A VODT droplet-forming eddy ranges from some location $y_0 < I$ to a location $y_0 + l > I$, which implies $l > I - y_0$ where. Based on the triplet-map definition, the eddy transfers 2/3 of the liquid interval $[y_0, I]$ to the droplet, while the rest remains in liquid form as defined in VODT. The implied surface recession is then $\varDelta y = \frac{2}{3}(I - y_0)$, corresponding to the LES-level volume conversion $\varDelta V = A_{s} \varDelta y$ where $A_{s}$ is the surface area of VOF interface element.

$\varDelta y$ is the only available length scale from which the droplet diameter D can be inferred, so $D = B \varDelta y$ is assumed, where B is a tunable parameter. Then the ODT droplet is deemed to represent N physical droplets, where $N = \varDelta V / \left( \frac{\pi }{6} D^3 \right)$ is based on assuming spherical droplets. Using $\varDelta V = A_{s}D/B$, this gives $N = 6A_{s} / (\pi BD^2)$. N need not to be an integer because it is meaningful only in terms of droplet statistics.

At the LES scale, it is assumed that the droplet spectrum in the gas phase is represented by a histogram based on either linear or geometric sized bins. Uniform linear bins of size dD are assumed here for illustration, although the reasoning is more general. To complete the formulation of VODT outputs, the total generation rate G(D) of droplets in the size range $[D, D + dD]$ is evaluated. To do this, the droplets number probability distribution, $g(D) = \frac{dG}{d D}$ per unit diameter increment is first evaluated. Based on the the results that follow, g(D) can be integrated over dD intervals to obtain the binned generation rates G(D).

The fixed VODT flow state is piecewise linear in u, where the slope discontinuity is determined by steady state momentum-flux balance at the phase interface, which is at the domain midpoint. The domain size, velocity difference across the domain, surface tension, and the phase viscosities and densities define a particular case.

The ODT eddy rate distribution $\lambda (y_0, l)$, which has no t dependence, has been evaluated exactly in closed from as a function of $y_0$ and l for a specified VODT state. To evaluate g(D), this distribution is integrated over its arguments subject to the constraint $D = B \varDelta y = \frac{2}{3}B(I - y_0)$, re-expressed as $y_0 = I - \frac{3}{2} \frac{D}{B}$. Formally this involves insertion of $\delta (y_0 - I + \frac{3}{2} \frac{D}{B})$ into the integral over $dy_0$, and thus

$$ \begin{aligned} g(D)&= N(D) \int _{\frac{3}{2} \frac{D}{B}}^{h - I + \frac{3}{2} \frac{D}{B}} dl \int dy_0 \, \lambda (y_0, l) \delta (y_0 - I + \frac{3}{2} \frac{D}{B})\\&= \frac{6A_{s}}{\pi BD^2} \int _{\frac{3}{2} \frac{D}{B}}^{h - I + \frac{3}{2} \frac{D}{B}} dl \, \lambda (I - \frac{3}{2} \frac{D}{B}, l). \end{aligned} $$

The moments of the drop number probability distribution are then defined by

$$ G_{i}(D) = \int _{D_{\min }}^{D_{\max }} g(D) \, D^{i}dD. $$

$G_{0}$ is the total number of droplets generated per unit time, $G_{1}$ is the total sum of diameter of the droplets per unit time, $\pi G_{2}$ is the total surface area of the droplets per unit time and $\pi G_{3}/6$ is the total volume of the droplets per unit time.

5 Results and Conclusion

As discussed, for this simple application VODT reduces to an algebraic system that is economical enough for on-the-fly runtime implementation. This makes VODT a computationally affordable tool to study different atomization processes. Figure 2 shows a normalized droplet atomization rate of interfacial breakup in different Reynolds and Weber numbers. The droplet generation rate, $G_{0}$, shows the total number of drops in time and is normalized by local shear $\varDelta u / h$. As seen in the Fig. 2, for relatively low injection velocities the atomization rate is primarily governed by the liquid surface tension. In Fig. 2 the density and viscosity ratios are both equal to 1. By varying the liquid/gas density ratio and keeping the viscosity ratio constant, Fig. 3 is generated, showing the atomization rate. As shown, VODT can generate the entire droplet dependent moments of the generation rate, $G_{i}(D)$ e.g. total volume of the drops per unit time for any flow condition.

Fig. 2

Normalized droplet generation rate [-]

Fig. 3

Normalized droplet generation rate at $\rho _{l}/\rho _{g} = 1, 10, 100$ with $\mu _{l}/\mu _{g} = 100$

As discussed earlier the main scope of this paper is to propose VODT as a subgrid model for LES/VOF simulations. Like ODT, VODT has three adjustable parameters C, Z, B that need to be calibrated for this planar shear layer application.

A possible basis for this that does not require external input is to run LES/VOF/VODT at different LES resolutions for the same case. If VODT is a good closure, then these results should all predict the same flow development and droplet statistics. As resolution improves, some of the droplet generation seen at the VODT level at coarse LES resolution should become LES-resolved, with VODT still giving the same results as at lower LES resolution for droplets still not resolved. This approach is currently under study and outcomes will be a part of discussions in future publications.