Filtered Navier-Stokes Equations Research Articles

Stochastic models in the under-resolved simulations of spray formation during high-speed liquid injection

In the under-resolved simulation of the high-speed liquid injection into stagnant air, the attention in subgrid-scale models is focused on events of intense gradients of the velocity in turbulent flow, i.e., on effects of intermittency. Three typical flows are considered—the in-nozzle flow, primary atomization zone, and secondary atomization of spray droplets. In the simulation of the first two flows, the filtered Navier–Stokes equations are forced by stochastic processes with properties which incorporate the statistical physics of fluid acceleration at the high Reynolds number—in this way we update the under-resolved acceleration. In the case of in-nozzle flow, the proposed stochastic subgrid acceleration model is combined with wall-damping function, and the ability in prediction of the velocity statistics is demonstrated. In the simulation of primary atomization, the approach with stochastic subgrid acceleration is combined with the volume of fluid method. This leads to intensification of the interface dynamics, resulting in additional corrugation, with more intense shearing and stretching of liquid structures are observed. Thereby, the experimental profiles of the time-averaged liquid volume fraction distribution for four different axial locations are rather well predicted. The secondary atomization of droplets is simulated by a new stochastic model for the breakup rate along the droplet path. To this end, the breakup rate is expressed as a function of turbulent viscous dissipation which evolves along the droplet path according to the proposed stochastic process. Preliminary assessment performed against the recent experiments shows the correct predictability of this model and its low sensibility to the grid density.

On wave-current interaction with a horizontal cylinder near free surface by a fourth-order-accurate finite volume compact solver

We present a numerical solver for dealing with the problem of wave-current interaction with a horizontal cylinder near free surface. The solver solves the filtered Navier-Stokes equations which are discretized using a fourth-order-accurate compact finite volume scheme on a collocated grid. The turbulence effect is modelled by a subgrid-scale model using a large eddy simulation, and the free surface and the cylinder are modelled by a volume of fluid method and a ghost cell immersed boundary method. The pressure-velocity coupling is ensured via a momentum interpolation method. We obtain the hydrodynamic force and its spectra characteristics, dynamic pressure distribution, streamline topology and vorticity patterns with different immersed depths under uniform currents, regular waves and wave-current flows. Validation is performed by comparing the calculated hydrodynamic force with the measured data within one wave period. The results show that the position of vortex shedding and dissipation is different for the cases under different immersed depth in regular waves and wave-current flows. The dynamic pressure around the cylinder as the force reaches its peak value changes with immersed depth. The value and phase of the force are sensitive to immersed depth.

Turbulence generation in the transitional wake flow behind a sphere

Turbulence generation in the transitional flow in the wake behind a sphere is studied with numerical simulations. The filtered Navier–Stokes equation and the large eddy simulation method are employed as the governing equation and the numerical method, respectively. The ΩR̃ vortex identification method is used to trace the evolution of vortices in the wake flow. The energy gradient theory is used to analyze the spike formation in the wake flow. The simulation results show that the vortex structure in the wake flow is the type of hairpin vortices, which is similar to that in a boundary layer flow. Ejection and sweep motions exist around the hairpin vortices. There are two most unstable regions in the wake where turbulence “burst” is first produced, one is near the center of the vortex head and the other is between the two vortex legs. There is a high-pressure zone above the vortex head due to the decrease in the streamwise velocity, and a soliton-like coherent structure exists in this area. The mechanism of turbulence generation in the wake is the discontinuity of the streamwise velocity, which makes the Navier–Stokes equation be singular. This singularity leads to the formation of the “negative spike” in the streamwise velocity. The amplitude of the “negative spike” reaches up to 60% of the incoming velocity, which is close to the situation in a boundary layer flow on a flat plate. It is concluded that the mechanism of turbulence generation in the wake flow is the same as that in the boundary layer flow.

A new ghost-cell/level-set method for three-dimensional flows

This study presents the implementation of a tailored ghost cell method in Hydro3D, an open-source large eddy simulation (LES) code for computational fluid dynamics based on the finite difference method. The former model for studying the interaction between an immersed object and the fluid flow is the immersed boundary method (IBM) which has been validated for a wide range of Reynolds number flows. However, it is challenging to ensure no-slip and zero gradient boundary conditions on the surface of an immersed body. In order to deal with this, a new sharp-interface ghost-cell method (GCM) is developed for Hydro3D. The code also employs a level-set method to capture the motion of the air-water interface and solves the spatially filtered Navier-Stokes equations in a Cartesian staggered grid with the fractional step method. Both the new GCM and IBM are compared in a single numerical framework. They are applied to simulate benchmark cases in order to validate the numerical results, which mainly comprise single-phase flow over infinite circular and square cylinders for low- and high-Reynolds number flows along with two-phase dam-break flows with a vertical cylinder, in which a good agreement is obtained with other numerical studies and laboratory experiments.

Toward coupled fire‐structure simulations for forecasting smoke leakage in case of concrete structures under fire

AbstractComputational modeling of the fire‐structure interaction demands the coupling between several models typically implemented in independent simulation software describing distinct physical processes. For instance, in the event of a fire breakout or building on fire, the fire not only increases the temperature of the immediate area but also initiates structural damage, leading to the spalling of the concrete along with the propagation of the smoke, heat, and radiation originating from the fire. As a result of the progressive structural damage caused by the fire such as macro cracks or breakthroughs in the wall, smoke and other hazardous gases begin to disperse and spread into previously unaffected zones. Thus, it represents a coupled multi‐physics problem, which is not well addressed in the scientific community, yet. Therefore, the investigation of such fire‐structure interactions requires a comprehensive computational modeling and simulation framework that couples the consequences of various phenomena with minimal invasive solver modifications. In this work, the structural model treats concrete as a multi‐phase porous material exposed to high temperatures resulting from the fire. For this purpose, the structural solver is at the moment manually coupled with a computational fluid dynamics (CFD) solver to setup a computational framework for such fire‐structure simulations. Herein, the combustion process of the fire and the smoke propagation is carried out using the open‐source software denoted as Fire Dynamics Simulator (FDS) and the thermo–hydro–mechanical fracture is computed using a FEniCS‐based solver. The finite‐difference method based FDS solves the filtered Navier–Stokes equations using the large‐eddy simulation technique to describe the turbulent flow and heat transfer including radiation and combustion inside the fluid domain. The concrete damage such as cracks, holes, or spalling is computed using a phase‐field method in a separate solver based on the thermal boundary conditions from the fire simulation. Subsequently, both solvers are coupled using the open‐source coupling framework preCICE. Based on the rate at which the crack or spalling is developing, the computational domain of the fluid solver has to be adapted taking the generated holes in the wall structure into account for the ongoing CFD simulation. It allows for the leakage of smoke gases through the concrete wall structure and thus the spreading of the fire scenario. The proposed model is illustrated by an exemplary case that forecasts the leakage of smoke and hazardous gases from the successive damage caused by the thermal spalling induced on a concrete wall structure.

Large-eddy simulation of airflow dynamics around a cluster of buildings

The wind flow around the buildings with different rooftops is studied numerically using large-eddy simulation (LES). The filtered Navier-Stokes equations in LES are used to compute the large eddies, whereas the dynamic Smagorinsky subgrid-scale (SGS) model calculates the small eddies. A turbulent spot method is applied for the synthesis of artificial turbulent fields at the inlet. In this separated and reattached flow, our aim is to analyze the wakes’ vortical structure along with the buildings of different heights and shapes. A large number of engineering applications involve precise predictions of the airflow around buildings to ensure performance and safety. To validate the numerical solver, a comparison is performed with the earlier reported numerical and experimental data. The turbulent flow characteristics are discussed in terms of instantaneous flow structure and time-averaged statistical flow quantities. All the cases are simulated for a Reynolds number [Formula: see text] to understand the turbulent airflow patterns. The flow shows a separable bubble at the leading edge of the rooftop of the building that leads to recirculation at the lee side of the first row of buildings. Due to the recirculation, the suction arises near the leading edge of the building to assure a continuous flow of air around the obstacles. Further, Reynold stresses show high momentum fluxes in the frontal region of the buildings.

Long Wave Induced Monopolar Vortex: Additional Large Eddy Simulation Case

An additional 3D large eddy simulation with a slightly different surface tension coefficient is carried out to study the detailed flow and turbulence characteristics of a long wave induced shallow water monopolar vortex. The numerical model adopted in this study is interDyMFoam, developed based on an OpenFOAM framework. The numerical model solves filtered Navier-Stokes equations for two immiscible fluids (air, water) with a standard Smagorinsky closure. In addition to the previous model results, newly obtained instantaneous model results are used to increase a number of realizations for ensemble averaging to reach more statistically converged results. The validation quality obtained in this study is shown to be similar to that from the previous simulation results in the low-pass filtered free surface elevation and horizontal component of velocity.

Three-dimensional DEM-CFD simulation of a lab-scale fluidized bed to support the development of two-fluid model approach

The present work is dedicated to the numerical study of the hydrodynamics of a pressurized fluidized-bed using an Euler–Lagrange approach, with the goal to gain insight into the Two-Fluid Model (TFM) approach. The gas phase is modeled by filtered Navier–Stokes equations, and the solid particles are tracked using a Discrete Element Method (DEM). Collisions are handled using a soft-sphere model. Numerical predictions of the mean (time-averaged) vertical particle velocity are compared with experimental measurements available from the literature, obtained from a Positron Emission Particle Tracking (PEPT) technique. In addition, DEM-Computational Fluid Dynamics (CFD) results are extensively compared with predictions from TFM numerical simulations. Results accounting for inelastic frictionless particle–particle collisions show a very good agreement with the experimental data and TFM results in the central zone of the reactor. In the near wall region the numerical simulation overestimates the downward particle velocity with respect to the experimental measurements, especially when the particle–wall friction is neglected. The influence of the friction at the wall is therefore further investigated and a local analysis of the particle–wall interactions is carried out. It is demonstrated that the long sustained contacts of particle assemblies with the wall in such a dense regime play a crucial role on the overall bed behavior. Therefore, it is recommended that this effect is taken into account in the boundary conditions of a TFM approach when it is used to predict bubbling fluidized beds. • DEM-CFD 3D simulation of a lab-scale fluidized bed • Comparison with experiments and Two-Fluid Model predictions • Local statistical analysis to characterize particle–wall interactions • Dominant friction effect at wall due to sustained particle–wall contacts

Investigation on multiscale features of cavitating flow in convergent-divergent test section using Eulerian–Lagrangian method

Cavitating flows in fluid machineries always present complicated phase morphologies including a wide range of interface length scales. In the present work, an Eulerian–Lagrangian cavitation model is proposed for simulating the unsteady cavitating flow and investigating the multiscale cavitating flow features in a convergent-divergent test section. Most importantly, a two-way transition algorithm is newly developed mainly based on the shape of resolved cavities and the relative size of microscale bubbles compared to the cells, for taking the transition between microscale discrete bubbles and macroscale vapor cavities into consideration. The transient flow field of the two-phase system is described by the filtered Navier–Stokes equation and the sub-grid scale model. The volume of fluid (VOF) method and a discrete bubble model (DBM) are combined for simulating resolvable water-vapor interfaces and unresolvable discrete bubbles, respectively. Applying the proposed multiscale modeling framework, the dissipated bubbles caused by the collapse of shedding cavities that are unable to be represented by the VOF method are well predicted. The results show that the VOF-DBM model is less sensitive to the mesh resolution compared to the VOF method and it is found the dissipated bubbles are transporting around the vortex core region. Detailed bubble dynamics including DBM to VOF and VOF to DBM transitions and the interaction between macroscale cavities and microscale bubbles are well presented, indicating that the proposed VOF-DBM model is a promising approach to reveal the multiscale cavitation characteristics within limited computing resources.

Large-eddy simulation of supercritical free-surface flow in an open-channel contraction

Large-eddy simulations (LES) of supercritical flow in a straight-wall, open-channel contraction of 6° and contraction ratio of 2:1 are performed. The LES code solves the filtered Navier-Stokes equations for two-phase flows (water-air) and employs the level-set method. The simulation was validated by replicating a previously reported experiment. Contours of the time-averaged velocities indicate that the flow loses energy and momentum in the contracting channel. Further, secondary currents in the contraction are redistributing momentum and are responsible for local up-and down-flows. The turbulent kinetic energy reaches very high values at the entrance of the contraction, mainly contributed by the streamwise normal stress. The flow contains coherent turbulence structures which are responsible for carrying low-momentum from the bed and the water surface towards the channel centre. Flow deceleration results in significant turbulence anisotropy in the contracted section. It is shown that mainly pressure drag contributes to the energy loss in the contraction.

Subgrid-scale models of isotropic turbulence need not produce energy backscatter

This investigation questions the importance of inverse interscale energy fluxes, the so-called energy backscatter, for the modelling of the energy cascade in large-eddy simulations (LES) of turbulent flows. The invariance of the filtered Navier–Stokes equations to divergence-preserving transformations of the subgrid-scale (SGS) stress tensor is exploited here to explore alternative representations of the local SGS energy fluxes. Numerical optimisation procedures are applied to the SGS stress tensor – obtained by filtering isotropic turbulence flow fields – to find alternative stresses that satisfy the filtered Navier–Stokes equations, but that produce negligible backscatter. These alternative SGS stresses show that backscatter represents not a flux of energy from the subgrid to the resolved scales, but conservative spatial fluxes, and that it need not be modelled to reproduce the local energetic exchange between the resolved and the subgrid scales in LES. From the perspective of statistical mechanics, it is argued that this is a consequence of the strong statistical irreversibility of inertial-range dynamics, which precludes inverse energy cascades even in a local sense. These findings show that the energy cascade is strongly unidirectional locally, and that it can be modelled as an irreversible sink of energy, justifying the extended use of purely dissipative SGS models in LES.

Tempered fractional LES modeling

The presence of non-local interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modelling framework for developing non-local LES subgrid-scale models, starting from the kinetic transport. We employ a tempered Lévy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\varDelta +\lambda )^{\alpha } (\cdot )$ , for $\alpha \in (0,1)$ , $\alpha \neq \frac {1}{2}$ and $\lambda >0$ in the filtered Navier–Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau, Phys. Fluids, vol. 6, issue 2, 1994, pp. 815–833). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model.

A priori analysis of subgrid-scale terms in compressible transcritical real gas flows

Of interest to the research community dealing with real gas flows, this study analyzes the influence of the physical complexity of real gases on the amplitude of subgrid-scale (SGS) terms present in the filtered Navier–Stokes equations to be solved in large eddy simulations. The direct numerical simulation results of three academic configurations (homogeneous isotropic turbulence, mixing layer, and channel flow) are filtered from the largest scale in the domain down to the Kolmogorov length scale. The analysis of the filtered flow variables consistently shows that the SGS turbulent stress and the SGS pressure cannot be neglected in the momentum equation. In the total energy equation, SGS pressure work and SGS internal and kinetic fluxes are found to be significant in the inertial zone of the turbulent kinetic energy spectrum. Since in the inertial zone, which corresponds to large filter sizes, specific models have not yet been designed for some of these terms, this study calls for such a modeling effort that will benefit the real gas and organic Rankine cycles research communities.

Time-averaging and temporal-filtering in wall-modeled large eddy simulation

A turbulent channel flow at a Reynolds number of Reτ=2000 is solved based on the spatially filtered Navier–Stokes equations using large eddy simulation and an in-house code. A nonequilibrium wall model is implemented to predict the flow in the wall layer based on the Reynolds-averaged approach. To mitigate the log-layer mismatch, which is often encountered in wall modeling, two temporal schemes are introduced to average the wall layer solution and to filter the flow information input to the wall layer. It is found that the time periods used for the time-averaging and temporal-filtering schemes affect the performance of the wall model. The results show that shorter time periods enable the wall model to respond to the flow structures in the outer layer and correctly predict the friction velocity. However, the prediction of the friction velocity also depends on the location of the matching point. Locating the matching point further from the wall results in better performance due to the compatibility of the subgrid scale model with the grid resolution further from the wall. The temporal-filtering scheme is used to remove nonessential high-frequency wavelengths that can disturb the functionality of wall modeling. Various combinations of the time-averaging and temporal-filtering time periods are investigated for different locations of the matching point. Overall, it is concluded that using a shorter period for time-averaging and a temporal-filtering period comparable to the turbulent diffusion timescale leads to improved results.

A Cartesian cut-cell based multiphase flow model for large-eddy simulation of three-dimensional wave-structure interaction

A multiphase flow numerical approach for performing large-eddy simulations of three-dimensional (3D) wave-structure interaction is presented in this study. The approach combines a volume-of-fluid method to capture the air-water interface and a Cartesian cut-cell method to deal with complex geometries. The filtered Navier–Stokes equations are discretised by the finite volume method with the PISO algorithm for velocity-pressure coupling and the dynamic Smagorinsky subgrid-scale model is used to compute the unresolved (subgrid) scales of turbulence. The versatility and robustness of the presented numerical approach are illustrated by applying it to solve various three-dimensional wave-structure interaction problems featuring complex geometries, such as a 3D travelling wave in a closed channel, a 3D solitary wave interacting with a vertical circular cylinder, a 3D solitary wave interacting with a horizontal thin plate, and a 3D focusing wave impacting on an FPSO-like structure. For all cases, convincing agreement between the numerical predictions and the corresponding experimental data and/or analytical or numerical solutions is obtained. In addition, for all cases, water surface profiles and turbulent vortical structures are presented and discussed.

DPM: A deep learning PDE augmentation method with application to large-eddy simulation

A framework is introduced that leverages known physics to reduce overfitting in machine learning for scientific applications. The partial differential equation (PDE) that expresses the physics is augmented with a neural network that uses available data to learn a description of the corresponding unknown or unrepresented physics. Training within this combined system corrects for missing, unknown, or erroneously represented physics, including discretization errors associated with the PDE's numerical solution. For optimization of the network within the PDE, an adjoint PDE is solved to provide high-dimensional gradients, and a stochastic adjoint method (SAM) further accelerates training. The approach is demonstrated for large-eddy simulation (LES) of turbulence. High-fidelity direct numerical simulations (DNS) of decaying isotropic turbulence provide the training data used to learn sub-filter-scale closures for the filtered Navier–Stokes equations. Out-of-sample comparisons show that the deep learning PDE method outperforms widely-used models, even for filter sizes so large that they become qualitatively incorrect. It also significantly outperforms the same neural network when a priori trained based on simple data mismatch, not accounting for the full PDE. Measures of discretization errors, which are well-known to be consequential in LES, point to the importance of the unified training formulation's design, which without modification corrects for them. For comparable accuracy, simulation runtime is significantly reduced. A relaxation of the typical discrete enforcement of the divergence-free constraint in the solver is also successful, instead allowing the DPM to approximately enforce incompressibility physics. Since the training loss function is not restricted to correspond directly to the closure to be learned, training can incorporate diverse data, including experimental data.

An adaptation of the immersed boundary method for turbulent flows over three-dimensional coastal/fluvial beds

A modification of the immersed boundary method is presented, using the discrete-forcing concept, for the large-eddy simulation of turbulent flows over coastal/fluvial beds. The method was adapted to be compatible with the usual surveying practice to obtain the bathymetry of coastal/fluvial beds, i.e., point measurements of water depth. An efficient fluid/solid point identification process and an improved trilinear interpolation algorithm were implemented for more accurate boundary forcing expressions and easier algorithm implementation. The temporal and spatial discretizations of the filtered Navier–Stokes equations were based on a time-splitting projection scheme and second-order finite-differences, respectively. The method was applied to the simulation of an oscillatory flow over bed ripples and a unidirectional flow over bed dunes. The accuracy of the computational results was verified by comparison to existing numerical and experimental results in the literature.

A three-dimensional Cartesian cut-cell/volume-of-fluid method for two-phase flows with moving bodies

A three-dimensional Cartesian cut-cell method for the large-eddy simulation of two-phase flows with moving bodies is presented in this study, which combines a volume-of-fluid method to capture the air-water interface and a moving body algorithm on a stationary, non-uniform, staggered, Cartesian grid. The filtered Navier–Stokes equations are discretised using the finite volume method with the PISO algorithm for velocity-pressure coupling and the dynamic Smagorinsky subgrid-scale model is employed to compute the effect of the unresolved (subgrid) scales of turbulence on the large scales. In the present study, the small cut-cells are unmodified and due to the use of an implicit time integration no instabilities occur during the computations. The versatility and robustness of the present two-phase flow model is illustrated via various two- and three-dimensional flow problems with fixed/moving bodies, such as dambreak flows with and without a square cylinder, a moving cylinder in a quiescent fluid, dambreak flow over a wet bed with a moving gate, water entry and exist of a circular cylinder, and landside-generated waves. Good agreement is obtained between the numerical results and the corresponding experimental measurements.

A fractional subgrid-scale model for turbulent flows: Theoretical formulation and a priori study

Coherent structures/motions in turbulence inherently give rise to intermittent signals with sharp peaks, heavy-skirt, and skewed distributions of velocity increments, highlighting the non-Gaussian nature of turbulence. This suggests that the spatial nonlocal interactions cannot be ruled out of the turbulence physics. Furthermore, filtering the Navier–Stokes equations in the large eddy simulation of turbulent flows would further enhance the existing nonlocality, emerging in the corresponding subgrid scale fluid motions. This urges the development of new nonlocal closure models, which respect the corresponding non-Gaussian statistics of the subgrid stochastic motions. To this end and starting from the filtered Boltzmann equation, we model the corresponding equilibrium distribution function with a Lévy-stable distribution, leading to the proposed fractional-order modeling of subgrid-scale stresses. We approximate the filtered equilibrium distribution function with a power-law term and derive the corresponding filtered Navier–Stokes equations. Subsequently in our functional modeling, the divergence of subgrid-scale stresses emerges as a single-parameter fractional Laplacian, (−Δ)α(·), α ∈ (0, 1], of the filtered velocity field. The only model parameter, i.e., the fractional exponent, appears to be strictly dependent on the filter-width and the flow Reynolds number. We furthermore explore the main physical and mathematical properties of the proposed model under a set of mild conditions. Finally, the introduced model undergoes a priori evaluations based on the direct numerical simulation database of forced and decaying homogeneous isotropic turbulent flows at relatively high and moderate Reynolds numbers, respectively. Such analysis provides a comparative study of predictability and performance of the proposed fractional model.

Influence of Diesel Fuel Viscosity on Cavitating Throttle Flow Simulations under Erosive Operation Conditions.

This work investigates the effect of liquid fuel viscosity, as specific by the European Committee for Standardization 2009 (European Norm) for all automotive fuels, on the predicted cavitating flow in micro-orifice flows. The wide range of viscosities allowed leads to a significant variation in orifice nominal Reynolds numbers for the same pressure drop across the orifice. This in turn, is found to affect flow detachment and the formation of large-scale vortices and microscale turbulence. A pressure-based compressible solver is used on the filtered Navier–Stokes equations using the multifluid approach; separate velocity fields are solved for each phase, which share a common pressure. The rates of evaporation and condensation are evaluated with a simplified model based on the Rayleigh–Plesset equation; the coherent structure model is adopted for the subgrid scale modeling in the momentum conservation equation. The test case simulated is a well-reported benchmark throttled flow channel geometry, referred to as “I-channel”; this has allowed for easy optical access for which flow visualization and laser-induced fluorescence measurements allowed for validation of the developed methodology. Despite its simplicity, the I-channel geometry is found to reproduce the most characteristic flow features prevailing in high-speed flows realized in cavitating fuel injectors. Subsequently, the effect of liquid viscosity on integral mass flow, velocity profiles, vapor cavity distribution, and pressure peaks indicating locations prone to cavitation erosion is reported.