A Kármán–Howarth–Monin equation for variable-density turbulence

We present a generalisation of the Kármán–Howarth–Monin (K–H–M) equation to include variable-density (VD) effects. The derived equation (i) reduces to the original K–H–M equation when density is a constant and (ii) leads to a VD analogue of the $4/5$-law with the same value of constant ($=4/5$) appearing as the prefactor of the dissipation rate. The equation is employed to understand negative turbulent kinetic energy production in a $\text{SF}_{6}$ turbulent round jet with an initial density ratio of 4.2. From a Reynolds-averaged Navier–Stokes (RANS) perspective, negative production means that the mean flow is strengthened at the expense of the energy of turbulent fluctuations. We show that the associated energy transfer is accomplished by the deformation of smaller turbulent eddies into large ones in the development region of the jet and is captured by the linear scale-by-scale energy transfer term in the VD K–H–M equation. The nonlinear transfer term of the VD K–H–M equation depicts a conventional forward cascade for all eddies having a size less than the Eulerian integral length scale, regardless of their orientation. The net effect is a retarded energy cascade in the non-Boussinesq jet that has not been accounted for by existing turbulence theories. Implications of this observation for turbulence modelling are discussed.

Spectral analysis of energy transfer in turbulent flows laden with heated particles

In this study we consider particle-laden turbulent flows with significant heat transfer between the two phases due to sustained heating of the particle phase. The sustained heat source can be due to particle heating via an external radiation source as in the particle-based solar receivers or an exothermic reaction in the particles. Our objective is to investigate the effects of fluid heating by a dispersed phase on the turbulence evolution. An important feature in such settings is the preferential clustering phenomenon which is responsible for non-uniform distribution of particles in the fluid medium. Particularly, when the ratio of particle inertial relaxation time to the turbulence time scale, namely the Stokes number, is of order unity, particle clustering is maximized, leading to thin regions of heat source similar to the flames in turbulent combustion. However, unlike turbulent combustion, a particle-laden system involves a wide range of clustering scales that is mainly controlled by particle Stokes number. To study these effects, we considered a decaying homogeneous isotropic turbulence laden with heated particles over a wide range of Stokes numbers. Using a low-Mach-number formulation for the fluid energy equation and a Lagrangian framework for particle tracking, we performed numerical simulations of this coupled system. We then applied a high-fidelity framework to perform spectral analysis of kinetic energy in a variable-density fluid. Our results indicate that particle heating can considerably influence the turbulence cascade. We show that the pressure-dilatation term introduces turbulent kinetic energy at a range of scales consistent with the scales observed in particle clusters. This energy is then transferred to high wavenumbers via the energy transfer term. For low and moderate levels of particle heating intensity, quantified by a parameter $\unicode[STIX]{x1D6FC}$ defined as the ratio of eddy time to mean temperature increase time, turbulence modification occurs primarily in the dilatational modes of the velocity field. However, as the heating intensity is increased, the energy transfer term converts energy from dilatational modes to divergence-free modes. Interestingly, as the heating intensity is increased, the net modification of turbulence by heating is observed dominantly in divergence-free modes; the portion of turbulence modification in dilatational modes diminishes with higher heating. Moreover, we show that modification of divergence-free modes is more pronounced at intermediate Stokes numbers corresponding to the maximum particle clustering. We also present the influence of heating intensity on the energy transfer term itself. This term crosses over from negative to positive values beyond a threshold wavenumber, showing the cascade of energy from large scales to small scales. The threshold is shown to shift to higher wavenumbers with increasing heating, indicating a growth in the total energy transfer from large scales to small scales. The fundamental energy transfer analysis presented in this paper provides insightful guidelines for subgrid-scale modelling and large-eddy simulation of heated particle-laden turbulence.

Revisiting the subgrid-scale Prandtl number for large-eddy simulation

The subgrid-scale (SGS) Prandtl number ($Pr$) is an important parameter in large-eddy simulation. Prior models often assume that the ‘$-5/3$’ inertial subrange scaling applies to the wavenumber range from 0 to $k_{\unicode[STIX]{x1D6E5}}$ (the wavenumber corresponding to the filter scale $\unicode[STIX]{x1D6E5}$) and yield a $Pr$ that is stability-independent and scale-invariant, which is inconsistent with experimental data and the results of dynamic models. In this study, the SGS Prandtl number is revisited by solving the co-spectral budgets of momentum and heat fluxes in an idealized but thermally stratified atmospheric surface layer. The SGS Prandtl number from the co-spectral budget model shows a strong dependence on the atmospheric stability and increases (decreases) as the atmosphere becomes stable (unstable), which is in good agreement with recent field experimental data. The dependence of $Pr$ on the filter scale is also captured by the co-spectral budget model: as the filter scale becomes smaller, the SGS Prandtl number decreases. Finally, the value of SGS Prandtl number under neutral conditions is shown to be caused by the dissimilarity between momentum and heat in the pressure decorrelation term and the flux transfer term. When the dissimilarity exists only in the flux transfer term, the fact that under neutral conditions the SGS Prandtl number is usually smaller than the turbulent Prandtl number for Reynolds-averaged Navier–Stokes simulations is an indication of a stronger spectral transfer coefficient for heat than for momentum. The model proposed in this study can be readily implemented.

Investigation of Pseudo Turbulent Scalar Transport in Two Phase Fluid Flow and Passive Scalar Mixing Using Simultaneous SPIV/PLIF

The presented work is focused on developing closure models for simulation of multiphase flow using multi-fluid models. In the two-fluid model, pseudo turbulent terms appear in both the heat transfer term in the energy equation and the mass transfer term in the species equation. These terms are often neglected due to lack of information, but recent studies show that they can indeed be significant in the simulation of the inter particle phenomena. In the present work, we experimentally investigate the importance of pseudo turbulent term in passive scalar transport. A simultaneous stereo particle image velocimetry and planar laser induced fluorescence (SPIV/PLIF) measurement of the field data for a liquid-solid flow is presented in this study. The results of this measurement are used to validate data from Particle Resolved DNS (PR-DNS) that in turn is used to develop the aforementioned closure models. In this work, results for a single sphere are presented for Reynolds number ranging from 50 to 150. In addition, results for arrays of spheres representing volume fractions of 0.1 and 0.2 are presented for the same range of Reynolds number.

Numerical Investigation of Metal-Foam Filled Liquid Piston Compressor Using a Two-Energy Equation Formulation Based on Experimentally Validated Models

The present study presents CFD simulations of a liquid-piston compressor with metal foam inserts. The term “liquid-piston” implies that the compression of the gas is done with a rising liquid-gas interface created by pumping liquid into the lower section of the compression chamber. The liquid-piston compressor is an essential part of a Compressed Air Energy Storage (CAES) system. The reason for inserting metal foam in the compressor is to reduce the temperature rise of the gas during compression, since a higher temperature rise leads to more input work being converted into internal energy, which is wasted during the storage period as the compressed gas cools. Liquid, gas, and solid coexist in the compression chamber. The two-energy equation model is used; the energy equations of the fluid mixture and the solid are coupled through an interfacial heat transfer term. The fluid mixture, which includes both the gas phase and the liquid phase, is modeled using the Volume of Fluid (VOF) method. Commercial CFD software, ANSYS FLUENT, is used, by applying its default VOF code, with user-defined functions to incorporate the two-energy equation formulation for porous media. The CFD simulation requires modeling of a negative momentum source term (drag), and an interfacial heat transfer term. The first one is the pressure drop due to the metal foam, which is obtained from experimental measurements. To obtain the interfacial heat transfer term, a compression experiment is done, which provides instantaneous pressure and volume data. These data are compared to solutions of a zero-dimensional compression model based on different heat transfer correlations from published references. By this comparison, a heat transfer correlation which is most suitable for the present study is chosen for use in the CFD simulation. The CFD simulations investigate two types of metal foam inserts, two different layouts of the insert (partial vs. full), and two different liquid piston speeds. The results show the influence of the metal foam inserts on secondary flows and temperature distributions.

Mechanical Work Potential

This paper considers the effect of heat transfer between fluid streams on the work output of a turbine. To correctly characterize the effect of heat transfer requires a new property, ‘mechanical work potential’, which is a measure of the maximum useful work that can be extracted from a fluid by an isentropic turbine exhausting to a fixed exit static pressure. A balance equation for the property, over a control volume, is developed. The equation shows that entropy creation through thermal mixing has no effect on turbine work. It does, however, show that a second heat transfer term, ‘thermal creation’, does alter turbine work. Thermal creation occurs in regions of the turbine where heat transfer occurs across a finite pressure difference. The term is the non-linear version of the acoustic energy creation term proposed by Lord Rayleigh in his thermo-acoustic criterion. The balance equation is then used to link local regions of thermal creation to changes in stage efficiency. The method is used to show that, in a modern high pressure turbine stage, heat transfer due to thermal mixing in the freestream causes a negligible change in efficiency and therefore can be ignored in the design process. The method is also used to show that heat transfer due to convective cooling results in ∼0.5% rise in stage efficiency. This is a significant and should be accounted for in the design process.

Modeling of Multisize Bubbly Flow and Application to the Simulation of Boiling Flows with the Neptune_CFD Code

This paper describes the modeling of boiling multisize bubbly flows and its application to the simulation of the DEBORA experiment. We follow the method proposed originally by Kamp, assuming a given mathematical expression for the bubble diameter pdf. The original model is completed by the addition of some new terms for vapor compressibility and phase change. The liquid-to-interface heat transfer term, which essentially determines the bubbles condensation rate in the DEBORA experiment, is also modeled with care. First numerical results realized with the Neptune_CFD code are presented and discussed.