Filtered Transport Equation Research Articles

The volume-filtering immersed boundary method

We present a novel framework to deal with static and moving immersed boundaries (IB) based on volume-filtering. In this strategy, called Volume-Filtering Immersed Boundary (VFIB) method, transport equations are derived by filtering the Navier-Stokes equations and accounting for stresses at the solid-fluid interface. The result is that boundary conditions that normally apply on the solid-fluid interface are transformed into bodyforces that apply on the right-hand side of the filtered transport equations. In this method, the filter width acts as a parameter that controls the level of resolution. The IB is considered well-resolved if the filter width is much smaller than the characteristic corrugation scale of the interface. There are several innovations in this IB method. First, it sheds light on the role of the internal flow which arises when the transport equations are solved inside the IB. We show that, it is essential to separate stresses due to the external and internal fluids in order to get accurate forces, and provide a method to do so. Second, we show that the volumes associated with Lagrangian forcing points on the boundary depend on the local topology of the surface. We provide a straightforward way to compute these volumes using a triangle tessellation of the interface and the surface density function. Third, we provide an efficient procedure to compute the solid volume fraction, thus, enabling tagging interior/exterior cells. This volume fraction is also involved in the procedure to separate stresses due to the external fluid from the total stresses. Fourth, we show a path forward to extend the VFIB method to Large Eddy Simulations involving IBs. Lastly, we apply the VFIB in several numerical tests involving two- and three-dimensional static and moving IBs. We show greatly improved results compared to prior IB methods. Further, we test several filter kernels and show that, for well-resolved IBs, the choice of the kernel plays little role.

A dynamic Spatially Averaged Two-Fluid Model for heat transport in moderately dense gas–particle flows

In this study, we derive a spatially averaged two-fluid model for heat transport in moderately dense gas–particle flows. In the context of multiphase turbulence modeling, closure models for the unresolved terms in the filtered transport equations in the presence of mesoscale heterogeneous particle clusters are postulated. In analogy to the drift velocity correction for the resolved gas–particle drag force, we propose to approximate the filtered interphase heat transfer by the resolved heat transfer corrected by a drift temperature. This drift temperature represents the gas-phase temperature fluctuations seen by the particles and can be expressed as a correlation between the solid volume fraction variations and the gas-phase temperature fluctuations, i.e., the turbulent internal energy. Therefore, transport equations for the turbulent internal energies of the phases are derived, where a cluster-induced turbulence production term arises in the gas-phase. Except for the interphase exchange terms, we find that closure models based on single phase turbulence modeling can be applied to the unresolved terms in the transport equations for both the filtered and turbulent internal energies. The interphase exchange terms can be expressed by the variances of the temperatures scaled by correlation coefficients. A dynamic adjustment of the correlation coefficients by using test-filters in coarse-grid simulations is proposed. In an a priori study, the developed closure models show good agreement with the predictions obtained by filtering fine-grid, two-fluid model simulation data of Geldart type A and B particles in three-dimensional wall-bounded fluidized beds.

An Enhanced <bold>Σ</bold> -Y Spray Atomization Model Accounting for Diffusion due to Drift-Flux Velocities

<div class="section abstract"><div class="htmlview paragraph">Spray modeling techniques have evolved from the classic DDM (Discrete Drops Method) approach, where the continuous liquid jet is discretized into “drops” or “parcels” till advanced spray models often based on Eulerian approaches. The former technique, although computationally efficient, is essentially inadequate in highly dense jets, as in the near nozzle region of compression ignition engines, while the latter could lead to extreme levels of computational effort when resolved interface capturing methods, such as VoF (Volume of Fluids) and LS (Level-Set) types, are used. However, in a typical engineering calculation, the mesh resolution is considerably coarser than in these high fidelity computations. If one presumes that these interfacial details are far smaller than the mesh size, smoothing features over at least one cell ultimately results in a diffuse-interface treatment in a Eulerian framework. Therefore, it is explained that currently the greatest interest is focused on the development of diffuse interface computational models which use a variable called surface density of the interface (Σ), solved using an averaged or filtered transport equation, to model the atomization process. However, in this type of approach a single-fluid formulation is used solving a single momentum equation but neglecting some features of the relative velocity between the liquid and the gas phases. This practice is not an issue for diesel sprays under nominal operating conditions of current engines, i.e., under high injection pressure and chamber density. Nevertheless, in certain combustion strategies, due to lower ambient densities (highly premixed combustion) or the usage of substitute fuels, the slip between phases becomes more important and must be fully considered. In this work, we propose a new developed Σ-Y Spray Atomization Model that accounts for diffusion due to Drift-Flux velocities, providing a predictive behavior under all engine conditions. In addition, the proposed formulation creates a mechanism by which the interfacial dynamics can impact the transport of the liquid mass fraction, therefore, making the interfacial density an active scalar, fully coupled with the rest of the flow, overcoming the limitations of previous formulations.</div></div>

Effects of radiative heat transfer on the turbulence structure in inert and reacting mixing layers

We use large-eddy simulation to study the interaction between turbulence and radiative heat transfer in low-speed inert and reacting plane temporal mixing layers. An explicit filtering scheme based on approximate deconvolution is applied to treat the closure problem arising from quadratic nonlinearities of the filtered transport equations. In the reacting case, the working fluid is a mixture of ideal gases where the low-speed stream consists of hydrogen and nitrogen and the high-speed stream consists of oxygen and nitrogen. Both streams are premixed in a way that the free-stream densities are the same and the stoichiometric mixture fraction is 0.3. The filtered heat release term is modelled using equilibrium chemistry. In the inert case, the low-speed stream consists of nitrogen at a temperature of 1000 K and the highspeed stream is pure water vapour of 2000 K, when radiation is turned off. Simulations assuming the gas mixtures as gray gases with artificially increased Planck mean absorption coefficients are performed in which the large-eddy simulation code and the radiation code PRISSMA are fully coupled. In both cases, radiative heat transfer is found to clearly affect fluctuations of thermodynamic variables, Reynolds stresses, and Reynolds stress budget terms like pressure-strain correlations. Source terms in the transport equation for the variance of temperature are used to explain the decrease of this variance in the reacting case and its increase in the inert case.

Entropy Filtered Density Function for Large Eddy Simulation of Turbulent Flows

A methodology termed entropy filtered density function (En-FDF) is developed to account for entropy transport in large eddy simulation of turbulent flows. The filtered entropy transport equation includes several unclosed terms which are the subgrid scale entropy flux and the filtered entropy generation resulting from irreversibilities in heat conduction, mass diffusion, chemical reaction and viscous dissipation. The En-FDF contains the complete statistical information about entropy, velocity, scalar and turbulent frequency fields and thus, provides closure for all the unclosed terms in the filtered transport equations. An exact transport equation is derived for the En-FDF which contains the effects of convection along with chemical reaction and its associated entropy generation in closed forms. The En-FDF transport is modeled by a set of stochastic differential equations. The numerical solution procedure is based on a hybrid finite difference/Monte Carlo method in which the large eddy simulation filtered transport equations are solved by the finite difference, and the stochastic differential equations are solved by a Lagrangian Monte Carlo procedure. This methodology is applied for large eddy simulation of a turbulent shear layer involving transport of passive scalars and predictions are compared against the data generated by direct numerical simulation. The En-FDF predictions show favorable agreement with the direct numerical simulation data.

E221 LESにおける局所的なエントロピー生成の予測手法に関する研究(一般講演(9))

LES is a promising tool for predicting local entropy production in turbulent flows. Currently, there is no numerical model available for industrial design with an acceptable cost and prediction accuracy. Therefore the purpose of this research is modeling the entropy production in turbulent heat transfer fields, based on the filtered transport equations for LES. The entropy production in turbulent flows occurs in two ways: viscous dissipation and heat conduction. In our earlier work, the production by viscous dissipation was in quantitative agreement, but the production by heat conduction was not. In this paper, we introduced improved models for the production by heat conduction. One of the three models that were considered in this paper shows a smaller difference between DNS and LES.

Filtered models for scalar transport in gas–particle flows

We employ a kinetic-theory based two-fluid model to develop a filtered two-fluid model for scalar transport in gas–particle flows. The filtering procedure gives rise to terms describing the filtered interphase heat/mass transfer and filtered scalar diffusion, which need to be constituted in order to close the filtered transport equations. In this work, the closure for these terms is accomplished by performing fine-grid simulations of the two-fluid model in a two-dimensional periodic domain. Filtered scalar diffusion is investigated by imposing a lateral mean gradient in the scalar for each phase. Interphase energy/mass transport is investigated by prescribing a heat/species source (sink) in the solids (gas) phase, such that the energy/species content of the mixture is preserved. The variation of the filtered transport coefficients with respect to filtered particle volume fraction, and scaling with respect to filter size, filtered scalar shear rate, and filtered slip velocity is discussed. We find the filtered interphase heat transfer coefficient to be as much as two orders of magnitude smaller than the microscopic interphase heat transfer coefficient. The model for filtered scalar diffusion is found to have a form very similar to that calculated for single phase turbulent flows. We also calculate the filtered Prandtl number for each phase.

Filtered density function simulator on unstructured meshes

A new computational filtered density function (FDF) methodology is developed for large eddy simulation (LES) of turbulent reacting flows. This methodology is based on a Lagrangian Monte Carlo (MC) FDF solver constructed on a domain portrayed by an unstructured mesh. The base filtered transport equations on this mesh are solved by a finite-volume (FV) method. The consistency of the hybrid FV-MC solver and the realizability of the simulated results are demonstrated via LES of a temporally developing mixing layer. The overall performance of the model is appraised by comparison with direct numerical simulation (DNS) data. The algorithmic implementation in the commercial software ANSYS-FLUENT facilitates future FDF-LES of turbulent combustion in complex configurations.

Frequency-velocity-scalar filtered mass density function for large eddy simulation of turbulent flows

A methodology termed “frequency-velocity-scalar filtered mass density function” (FVS-FMDF) is developed for large eddy simulation (LES) of turbulent flows. The FVS-FMDF takes account of unresolved subgrid scales (SGSs) by considering the joint probability density function (PDF) of the frequency, the velocity, and the scalar fields. An exact transport equation is derived for the FVS-FMDF in which the effects of convection and chemical reaction are in closed forms. The unclosed terms in this equation are modeled in a fashion similar to PDF methods in Reynolds-averaged Navier–Stokes simulations. The FVS-FMDF transport is modeled via a set of stochastic differential equations (SDEs). The numerical solution procedure is based on a hybrid finite-difference (FD)/Monte Carlo (MC) method in which the LES filtered transport equations are solved by the FD, and the set of SDEs is solved by a Lagrangian MC procedure. LES of a temporally developing mixing layer is conducted via the FVS-FMDF, and the results are compared with those via the Smagorinsky SGS closure. All these results are also assessed by comparison with those obtained by direct numerical simulation (DNS). The FVS-FMDF predictions show favorable agreements with DNS data.

The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction

A formulation of the approximate deconvolution model (ADM) for the large-eddy simulation (LES) of compressible flows in complex geometries is detailed. The model is applied to supersonic compression ramp flow where shock-turbulence interaction occurs. With the ADM approach an approximation to the unfiltered solution is obtained from the filtered solution by a series expansion involving repeated filtering. Given a sufficiently good approximation of the unfiltered solution at a time instant, the flux terms of the underlying filtered transport equations can be computed directly, avoiding the need to explicitly compute subgrid-scale closures. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation and a dynamically estimated relaxation parameter. Results of the large-eddy simulation of the turbulent supersonic boundary layer along a compression ramp compare well with filtered DNS data. The filtered shock solution is correctly predicted by the ADM procedure, demonstrating that turbulent and nonturbulent subgrid-scales are properly modeled. We found that a computationally expensive shock-capturing technique was not necessary for stable integration. As a consequence, the computational effort for simulations with ADM is approximately as large as for a coarse-grid DNS with a hybrid compact-upwind-ENO scheme, since the additional computational cost for the subgrid-scale model is more than compensated due to the fact that in the LES flux-derivatives can be computed by linear central finite differences on the entire domain.

Large eddy simulation of a turbulent reacting jet with conditional source-term estimation

The Conditional Source-term Estimation (CSE) method was recently proposed to close the chemical source terms occurring in the spatially filtered transport equations of species and enthalpy for Large Eddy Simulation (LES) of nonpremixed reacting flows [W. K. Bushe and H. Steiner, Phys. Fluids 11, 1896 (1999)]. The model is based on the Conditional Moment Closure hypothesis, which provides fairly accurate predictions for the conditional averages of the chemical reaction rates as functions of the conditionally averaged composition vector and temperature with the mixture fraction being an appropriate conditioning variable. In CSE the conditionally averaged composition vector and temperature are obtained by mapping the corresponding spatially filtered scalar fields resolved by the LES into the conditioning (i.e., mixture fraction) space. After the conditional averages of the chemical reaction rates are approximated in mixture fraction space, these are mapped into the physical space to close the source terms in the LES transport equations for the reactive scalars. The present simulation of a turbulent reacting jet is the first test of this new closure in a self-sustained predictive LES. A two-step reduced chemical kinetic mechanism for methane–air flames was used. The results of the simulation, which are in reasonable agreement with available experimental data, prove the model’s predictive capabilities as well as its robustness and feasibility for LES.

(2-22) Numerical Analysis of the Interaction between Thermo-Fluid Dynamics and Auto-Ignition Reaction in Spark Ignition Engines((SI-7)S. I. Engine Combustion 7-Modeling)

The interaction between thermo-fluid dynamics and auto-ignition reaction was analyzed by the CFD simulation integrating the low temperature oxidation reaction. In the first analysis, a compression auto-ignition of a premixed mixture in a rapid compression machine was solved in a laminar axisymmetric field. The computed results show that the temperature field becomes non-uniform at the end of compression, being higher in the outer region and lower in the central region ana near wan regions, This is due to the vortex induced in the piston-liner corner and wall heat transfer. It is shown that the auto-ignition initiates different places depending on whether local temperatures are outside or inside the negative temperature coefficient regime of the low temperature oxidation reaction. In the second, the auto-ignition in an s. i. engine was analyzed in a three dimensional turbulent field by solving the spatially filtered transport equations instead of the Reynolds averaged ones in consideration of the strong non-linearity to temperature in the reaction terms in the fluctuating temperature field. The computed results for a disk-shaped combustion chamber with a central spark plug show that temperature fluctuations appear during compression due to random variations in the wall heat transfer which were generated by the random motion flow, though a uniform temperature field was assumed as the initial condition. Figure 1 shows the computed pressure compared with the experimental results. The calculation terminated at the auto-ignition initiation, which was assumed as the time when any local temperature of the unburned mixture reached 1100 K. The predicted time of the first local auto-ignition appearance is seen to be 1.1 deg. BTDC, which falls in the range of the cycle variation in experimental results. The computed temperature fields of the unburned mixture at this time are shown in Figure 2. The figure demonstrates that the auto-ignition sites appear sporadically as seen in the high-speed photographs presented in the literature.

Large-Eddy Simulation of a Turbulent Hydrogen Diffusion Flame

In this work a large-eddy simulation (LES) of a turbulent hydrogen jet diffusion flame is presented. The numerical method handles fluctuations of density in space and in time, but assumes density to be independent of pressure (incompressibility). The chemical composition of the fluid is described by solving the filtered transport equation for mixture fraction f. Density, viscosity and temperature are evaluated assuming chemical equilibrium. To account for sub-grid fluctuations of f, its sub-grid distribution is presumed to have the shape of a β-function. The results of the simulation are discussed extensively. The influence of inlet boundary conditions is addressed and radial profiles at different axial positions are shown for a complete set of one-point statistical data. Agreement of numerical results and experimental data is very good. Furthermore, a comparison of Reynolds- and Favre-averages is done and energy spectra at different locations in the flame are discussed.