Direct Numerical Simulation Approach Research Articles

Conjugate heat transfer analysis of rectangular cooling channels using modeled and direct numerical simulation of turbulence

We carry out a computational study of conjugate heat transfer in rectangular cooling channels typical of liquid propellant rocket engines, with the goal of establishing the predictive capability of Reynolds averaged Navier-Stokes (RANS) solvers. A classical Spalart-Allmaras RANS solver is here used, which is iteratively coupled with a Fourier solver which solves for conduction within the duct walls. Comparison is made with the reference solutions of the same problem obtained with a direct numerical simulation (DNS) approach, which allows to describe the full features of flow and heat evolution in the channel. Numerical simulations are carried out for different values of the fluid-solid thermal conductivity ratio to bring out conjugate heat transfer effects. Finite conductivity of the solid is found to imply reduced thermal efficiency of the overall system as a result of both increased thermal resistance, and asymmetric heat loading on the fluid. We find that, despite the absence of the secondary motions, the RANS/Fourier solver can accurately estimate the pressure drop. Differences in the prediction of thermal effects are generally larger. However, the wall temperature distributions are predicted by RANS with reasonable accuracy, and discrepancy in the overall heat transfer coefficient is no larger than 10%.

Direct numerical simulation of flapping flags in grid-induced turbulence

A fully resolved direct numerical simulation approach for investigating flexible bodies forced by a turbulent incoming flow is designed to study the flapping motion of a flexible flag at the moderate Reynolds number. The incoming turbulent flow is generated by placing a passive grid at the inlet of the numerical domain and the turbulence level of the flow, impacting that the flag can be controlled by changing its downstream distance from the grid. The computational framework is based on the immersed boundary method for dealing with arbitrary geometries and implemented using a graphics-processing-unit accelerated parallelization to increase the computational efficiency. The grid-induced turbulent flow is first characterized by means of comparison with well-known results for decaying turbulence and a scale-by-scale analysis. Then, the flag-in-the-wind problem is revisited by exploring the effect of the turbulence intensity on self-sustained flapping. While the latter is still manifesting under strong fluctuations, the main features of the oscillation (including its amplitude and frequency) are altered by turbulence, whose fingerprint can also be qualitatively detected by spectral analysis. Besides their relevance for advancing the fundamental understanding of the fluid–structure interaction in turbulence, these findings have potential impact for related applications, e.g., aeroelastic energy harvesting or flow control techniques.

Effective control of aeolian tone using a pair of splitter plates

We propose here a novel arrangement consisting of a pair of splitter plates kept close to a circular cylinder to suppress aeolian tone. Splitter plates have a circular arc shape and are located concentrically close to the top and the bottom surfaces of the cylinder. Solutions for flow past a cylinder and splitter plates assembly have been obtained by solving unsteady, two-dimensional, compressible, Navier-Stokes equations using the direct numerical simulation (DNS) approach. Simulations have been performed for the free-stream Reynolds number of Re=150 and the Mach number M=0.2. Shear layers separated from the splitter plates and the circular cylinder interacted with each other. For certain positions of the splitter plates, considerable reductions in flow and sound disturbances have been noted. Maximum reduction in the net radiated sound power obtained by inserting splitter plates is around 21 dB as compared to the no-splitter plate case. Sound fields for such cases with significant reduction in the radiated sound, display the quadrupole behavior instead of the dipolar behavior which is usually observed for a laminar flow past a bluff body.

Tail shapes lead to different propulsive mechanisms in the body/caudal fin undulation of fish

In this paper, we investigate the hydrodynamics of swimmers with three caudal fins: a round one corresponding to snakehead fish ( Channidae), an indented one corresponding to saithe ( Pollachius virens), and a lunate one corresponding to tuna ( Thunnus thynnus). A direct numerical simulation (DNS) approach with a self-propelled fish model was adopted. The simulation results show that the caudal fin transitions from a pushing/suction combined propulsive mechanism to a suction-dominated propulsive mechanism with increasing aspect ratio ( AR). Interestingly, different from a previous finding that suction-based propulsion leads to high efficiency in animal swimming, this study shows that the utilization of suction-based propulsion by a high- AR caudal fin reduces swimming efficiency. Therefore, the suction-based propulsive mechanism does not necessarily lead to high efficiency, while other factors might play a role. Further analysis shows that the large lateral momentum transferred to the flow due to the high depth of the high- AR caudal fin leads to the lowest efficiency despite the most significant suction.

Numerical Simulation of Instabilities in the Boundary-Layer Transition Experiment Flowfield

The development of disturbances in the boundary-layer transition (BoLT) flight experiment flowfield is investigated using a recently developed “quiet direct numerical simulation (DNS)” approach. By using a combination of low-dissipation numerics and a novel shock capturing method, numerical noise is significantly reduced, enabling the simulation and analysis of early stages of the transition process that are governed by the linear growth of small-amplitude disturbances. The freestream conditions of the present simulations correspond to experiments conducted in the Purdue Boeing/U.S. Air Force Office of Scientific Research Mach 6 quiet tunnel under quiet flow at a unit Reynolds number of . First, the steady-state flowfield is computed and then compared to infrared images, demonstrating excellent agreement with experimentally measured steady streamwise wall heat flux streaks. Next, an unsteady numerical simulation is performed with continuous stochastic forcing to excite boundary-layer instabilities. Two-dimensional time-series snapshots of the disturbance flowfield are saved at several streamwise locations. Sparsity-promoting dynamic mode decomposition (SPDMD) is used to extract dominant modes from the snapshot sequences. The frequencies of dominant modes are compared to power spectral densities obtained from wall pressure fluctuations measured in the experiments. Based on numerical and experimental data, three distinct instabilities in the BoLT flowfield are identified. The quiet DNS approach used in conjunction with controlled disturbances to excite instabilities, as well as SPDMD to analyze them, represents a new high-fidelity method of investigating stability and transition in complex three-dimensional flowfields.

Impact of aerosols and turbulence on cloud droplet growth: an in-cloud seeding case study using a parcel–DNS (direct numerical simulation) approach

Abstract. This paper investigates the relative importance of turbulence and aerosol effects on the broadening of the droplet size distribution (DSD) during the early stage of cloud and raindrop formation. A parcel–DNS (direct numerical simulation) hybrid approach is developed to seamlessly simulate the evolution of cloud droplets in an ascending cloud parcel. The results show that turbulence and cloud condensation nuclei (CCN) hygroscopicity are key to the efficient formation of large droplets. The ultragiant aerosols can quickly form embryonic drizzle drops and thus determine the onset time of autoconversion. However, due to their scarcity in natural clouds, their contribution to the total mass of drizzle drops is insignificant. In the meantime, turbulence sustains the formation of large droplets by effectively accelerating the collisions of small droplets. The DSD broadening through turbulent collisions is significant and therefore yields a higher autoconversion rate compared to that in a nonturbulent case. It is argued that the level of autoconversion is heavily determined by turbulence intensity. This paper also presents an in-cloud seeding scenario designed to scrutinize the effect of aerosols in terms of number concentration and size. It is found that seeding more aerosols leads to higher competition for water vapor, reduces the mean droplet radius, and therefore slows down the autoconversion rate. On the other hand, increasing the seeding particle size can buffer such a negative feedback. Despite the fact that the autoconversion rate is prominently altered by turbulence and seeding, bulk variables such as liquid water content (LWC) stays nearly identical among all cases. Additionally, the lowest autoconversion rate is not co-located with the smallest mean droplet radius. The finding indicates that the traditional Kessler-type or Sundqvist-type autoconversion parameterizations, which depend on the LWC or mean radius, cannot capture the drizzle formation process very well. Properties related to the width or the shape of the DSD are also needed, suggesting that the scheme of Berry and Reinhardt (1974) is conceptually better. It is also suggested that a turbulence-dependent relative-dispersion parameter should be considered.

Binary collision of CMAS droplets—Part I: Equal-sized droplets

Abstract

Efficacy of the FDA nozzle benchmark and the lattice Boltzmann method for the analysis of biomedical flows in transitional regime

Flows through medical devices as well as in anatomical vessels despite being at moderate Reynolds number may exhibit transitional or even turbulent character. In order to validate numerical methods and codes used for biomedical flow computations, the US Food and Drug Administration (FDA) established an experimental benchmark, which was a pipe with gradual contraction and sudden expansion representing a nozzle. The experimental results for various Reynolds numbers ranging from 500 to 6500 were publicly released. Previous and recent computational investigations of flow in the FDA nozzle found limitations in various CFD approaches and some even questioned the adequacy of the benchmark itself. This communication reports the results of a lattice Boltzmann method (LBM) – based direct numerical simulation (DNS) approach applied to the FDA nozzle benchmark for transitional cases of Reynolds numbers 2000 and 3500. The goal is to evaluate if a simple off the shelf LBM would predict the experimental results without the use of complex models or synthetic turbulence at the inflow. LBM computations with various spatial and temporal resolutions are performed—in the extremities of 45 million to 2.88 billion lattice cells—executed respectively on 32 CPU cores of a desktop to more than 300,000 cores of a modern supercomputer to explore and characterize miniscule flow details and quantify Kolmogorov scales. The LBM simulations transition to turbulence at a Reynolds number 2000 like the FDA’s experiments and acceptable agreement in jet breakdown locations, average velocity, shear stress, and pressure is found for both the Reynolds numbers.Graphical A bisecting plane showing the FDA nozzle and vorticity magnitude at t = 10 s for throat Reynolds numbers of 2000 and 3500

A Novel Approach for Direct Numerical Simulation of Hydraulic Fracture Problems

Hydraulic fracturing is a non-linear and multi-physics problem involving the break up of a solid medium due to the action of hydrodynamic forces. Fluid and solid mechanics are involved at the same time together with fracture mechanics. Despite its relevance in many scientific and engineering fields, the theoretical and numerical description of hydraulic fracturing remains a challenging matter and the capabilities of existing models for applications are still limited. In this context, we propose a novel numerical approach to the Direct Numerical Simulation of hydraulic fracturing based on the Navier–Stokes equations coupled with peridynamic theory of solid mechanics through a multi-direct Immersed Boundary Method. The main advantage of this approach consists in the reliable crack-detection and tracking capabilities of peridynamics together with the capability of the Immersed Boundary Method of managing no-slip and no-penetration boundary conditions on complex and time-evolving interfaces. A massive-parallel solver based on this model has been implemented. We present a detailed theoretical description of the proposed methodology as well as the results of an extensive validation campaign for the new solver. Different benchmarking tests are provided together with the qualitative results of a simulation reproducing the fracture of a solid structure in a laminar, unsteady flow.

Prediction of the aerodynamic sound generated due to flow over a cylinder performing combined steady rotation and rotary oscillations.

Analysis of sound generated due to a laminar flow past a circular cylinder subjected to the mean rotation along with the rotary oscillating motion has been performed for the Reynolds number Re = 150 and the Mach number M = 0.2. The direct numerical simulation approach has been used to study modifications in the generated sound field over a range of forcing parameters using disturbance pressure field information. Flow and sound fields are accurately resolved over a nondimensional radial distance r≤100 from the center of the cylinder. Frequencies, as well as wavelengths of generated sound waves, have been effectively altered by varying the forcing frequency-ratio, whereas the directivity nature of the radiated sound field has been modified by varying the forcing amplitude-ratio. Doak's decomposition technique has been used to understand the reasons behind changes in the radiated sound fields as the forcing parameters are varied.

Direct numerical simulation of a tip-leakage flow in a planar duct with a longitudinal slit

A planar duct flow configuration with a cross-flow injected from a longitudinal slit close to the upper wall of the duct is studied by using a direct numerical simulation approach to explore the underlying flow mechanism in relation to the tip-leakage vortex (TLV), which is one of the most important flow phenomena in turbomachinery. Major characteristics of TLV in a rotor of turbomachinery are identified in the current flow model. The analysis of mean and instantaneous flow fields reveals that the interaction between the main (axial) flow and jet (cross) flow is the primary source of the generation of the TLV. The evolution of the TLV is then investigated, and a vortex breakup phenomenon is identified. The evolution of TLV can be divided into three phases, i.e., the formation phase, the breakup phase, and the diffusion phase. Mean streamlines and turbulence kinetic energy (TKE) budgets are analyzed, showing that the high TKE central spot in the formation phase is due to the interaction between highly swirling vortex filaments and mean velocity gradient. In the outer part of the TLV, the TKE is mainly produced in the shear-layer and transported toward the center by the turbulence transport.

Two-phase flow of CO2-brine in a heterogeneous sandstone: Characterization of the rock and comparison of the lattice-Boltzmann, pore-network, and direct numerical simulation methods

Understanding the physics of two-phase flow of CO2 and brine in porous geological formations is essential to sequestration of carbon dioxide in deep saline reservoirs, as well as the older problem of enhanced oil recovery from hydrocarbon reservoirs by CO2 injection. A pilot CO2 injection in Decatur, Illinois, was undertaken, with the injection zone being the highly saline and heterogeneous Mt. Simon sandstone, in order to better understand the feasibility of full-scale sequestration process. This paper reports the results of an extensive study of the morphology of the sandstone and its heterogeneity, and simulation of single-phase and two-phase flow of CO2 and brine in the formation's three-dimensional images. As we demonstrate by extensive analysis, the formation is much more heterogeneous than the typical sandstone, such as Berea sandstone. In addition to characterizing the morphology of the sandstone and computing its important flow characteristics, an important goal of the study is to compare the accuracy and computational efficiency of three distinct simulation approaches, namely, the lattice-Boltzmann (LB) approach, direct numerical simulation (DNS) of the governing equations of fluid flow that uses the finite-volume method coupled with the OpenFOAM simulator, and pore-network (PN) simulation. After validating the simulators by comparing the computed relative permeabilities that they produce for Berea sandstone, we simulate displacement of brine by CO2 at low and relatively high capillary numbers, and compute the relative permeabilities and other quantities of interest. We demonstrate that all the three methods provide consistent relative permeability-saturation functions that are in close agreement with one another. However, although the LB and DNS both produce similar relative permeabilities, the DNS approach is computationally more efficient because it simulates drainage by only a single set of computations over the entire saturation range, whereas the LB simulation requires separate simulation for each set saturation. Thus, the question of what method to use for simulating such flow processes at the scale of core plugs should mainly be addressed based on the computational time that one can afford and the computational resources that one has access to. Another important question addressed is the effect of the resolution of the computational grids or lattices used, particularly when one uses the LB method with voxelized images of porous media. We show that, unlike many claims in the past, one may need many lattice units per voxel in order to obtain reliable, lattice-independent results.

Numerical modelling of wall-bounded turbulent flows based on the CABARET scheme

In this study, the classical problem of turbulent plane channel flow at Rem = 13750 and Rem = 21900 is investigated using a numerical algorithm based on the CABARET scheme. Simulations are performed on the fine grids, resolving all the spatial scales of the turbulence according to a direct numerical simulation approach, as well as on coarse grids standard for large eddy simulation of wall-bounded turbulent flows. In the latter case, in order to obtain a more accurate representation of the momentum transfer towards the wall, artificial boundary conditions are introduced. It allows modelling with high accuracy the mean flow characteristics. Numerical results obtained by the CABARET scheme are compared with numerical results obtained by pseudospectral method.

Direct numerical simulation and statistical analysis of stress-driven turbulent Couette flow with a free-slip boundary

The effects of free-slip boundary on shear turbulence are studied numerically using the direct numerical simulation (DNS) approach. The flow considered in this study is a stress-driven turbulent Couette flow between two flat boundaries. The top boundary has an imposed shear stress in the streamwise direction and a free-slip condition for the streamwise velocity fluctuation and the spanwise velocity, and the bottom boundary satisfies the no-slip condition. This type of flow has a mean flow pattern similar to the turbulent plane Couette flow between a stationary flat plate and a moving flat plate but exhibits considerable differences in turbulence statistics due to the effects of the free-slip boundary. Statistical analysis based on the DNS data and theoretical derivation based on Taylor series expansion show that near the free-slip surface the turbulence variances of the three velocity components vary as quadratic functions of the vertical distance from the boundary while the Reynolds shear stress exhibits a linear behavior, which are very different from the counterparts near the no-slip boundary. The free-slip surface condition also leads to zero horizontal vorticities at the surface but allows nonzero vertical vorticity in the meantime, leading to considerable differences in the near-boundary statistics of vorticities and coherent vortex structures. Comparison of three DNS runs with different grid resolutions shows that smoothly resolving the more energetic turbulent flow structures near the free-slip boundary requires a higher horizontal grid resolution than that used for resolving the structures near the no-slip boundary.

High-Frequency Fluctuations in Post-stenotic Patient Specific Carotid Stenosis Fluid Dynamics: A Computational Fluid Dynamics Strategy Study

PurposeScreening of asymptomatic carotid stenoses is performed by auscultation of the carotid bruit, but the sensitivity is poor. Instead, it has been suggested to detect carotid bruit as neck’s skin vibrations. We here take a first step towards a computational fluid dynamics proof-of-concept study, and investigate the robustness of our numerical approach for capturing high-frequent fluctuations in the post-stenotic flow. The aim was to find an ideal solution strategy from a pragmatic point of view, balancing accuracy with computational cost comparing an under-resolved direct numerical simulation (DNS) approach vs. three common large eddy simulation (LES) models (static/dynamic Smagorinsky and Sigma).MethodWe found a reference solution by performing a spatial and temporal refinement study of a stenosed carotid bifurcation with constant flow rate. The reference solution left( {Delta x = 1.92 times 10^{ - 4} ;{text{m}},; Delta t = 5 times 10^{ - 5} ;{text{s}}} right) was compared against LES for both a constant and pulsatile flow.ResultsOnly the Sigma and Dynamic Smagorinsky models were able to replicate the flow field of the reference solution for a pulsatile simulation, however the computational cost of the Sigma model was lower. However, none of the sub-grid scale models were able to replicate the high-frequent flow in the peak-systolic constant flow rate simulations, which had a higher mean Reynolds number.ConclusionsThe Sigma model was the best combination between accuracy and cost for simulating the pulsatile post-stenotic flow field, whereas for the constant flow rate, the under-resolved DNS approach was better. These results can be used as a reference for future studies investigating high-frequent flow features.

Multiscale simulation of elongated particles in fluidised beds

In this paper, we present a number of key numerical methods that can be used to study elongated particles in fluid flows, with a specific emphasis on fluidised beds. Fluidised beds are frequently used for the production of biofuels, bioenergy, and other products from biomass particles, which often have an approximate elongated shape. This raises numerous issues in a numerical approach such as particle-particle contact detection and the accurate description of the various hydrodynamic forces, such as drag, lift, and torque, that elongated particles experience when moving in a fluid flow. The modelling is further complicated by a separation of length scales where industrial flow structures that can extend for many metres evolve subject to solid-solid and solid-fluid interactions at the millimetre scale. As a result, it is impossible to simulate both length scales using the same numerical approach, and a multiscale approach is necessary. First, we outline the direct numerical simulation (DNS) approach that may be employed to estimate hydrodynamic force closures for elongated particles in a fluid flow. We then describe the key aspects of a CFD-DEM approach, which can be used to simulate laboratory scale fluidisation processes, that must be addressed to study elongated particles. Finally, we briefly consider how current industrial-scale models, which concretely assume particle sphericity, could be adapted for the simulation of large collections of elongated particles subject to fluidisation.

Analysis of sound generation by flow past a circular cylinder performing rotary oscillations using direct simulation approach

Generation of sound due to laminar flow past a circular cylinder performing rotary oscillations has been studied using a direct numerical simulation approach. Two-dimensional, unsteady, compressible Navier-Stokes equations are directly solved using high resolution, physical dispersion relation preserving schemes. In this work, modifications in the flow induced acoustic noise due to imposed rotary oscillations have been discussed in detail. Simulations have been performed for a Reynolds number Re = 150 and a Mach number M = 0.2 over a wide range of forcing frequencies and amplitudes of rotary oscillation, specifically in the synchronization region. Rotary oscillating motion of a cylinder modifies the vortex shedding patterns in the wake region as compared to the case of flow past a stationary cylinder. The frequency and strength of shed vortices determine the nature of aerodynamic forces acting on the cylinder as well as sound generation. Reduction in sound generation has been observed for some of the forced oscillation cases as compared to the flow past a stationary cylinder case. The Doak’s decomposition methodology has been used to segregate the acoustic and hydrodynamic modes from the momentum density field to understand changes in the radiated sound field for different forcing conditions. Furthermore, disturbance pressure fields have been decomposed into a number of modes based on their significance, using a proper orthogonal decomposition (POD) technique in order to identify and quantify the contribution of the lift and drag dipoles to the sound field. In addition, POD modes of disturbance vorticity fields as well as noise source structures based on approximate Lighthill’s stress tensor are also obtained and related to the generated sound fields. This analysis concludes that the frequency of rotary oscillation dictates the frequency content of the flow induced sound field. Low frequency rotary oscillations trigger sound waves with low frequencies and large wavelengths. As the forcing frequency increases, the corresponding sound field displays shorter wavelengths. Directivity of the sound field is affected by the amplitude of rotary oscillation. A case with higher forcing amplitude distributes sound energy more evenly in all directions as compared to a lower forcing amplitude case. Prescription of rotary oscillations to the circular cylinder significantly alters the frequency, amplitude, and directivity of the generated sound field.

Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. I. Linear stability theory

There has been renewed interest in supersonic modes in hypersonic boundary layers, which have been previously thought to be insignificant due to their smaller amplitudes than Mack’s traditional second mode. Supersonic modes are associated with an unstable second mode synchronizing with the slow acoustic spectrum, causing sound to radiate outwards from the boundary layer. Because supersonic modes have not been observed experimentally, the majority of previous investigations either relied on Linear Stability Theory (LST) to study supersonic modes on a flat plate or observed them in the context of other research objectives. This two-part study uses a combined LST and Direct Numerical Simulation (DNS) approach to investigate the mechanism of supersonic modes in Mach 5 flow over a blunt cold-wall cone with thermochemical nonequilibrium effects. Paper I focuses on LST with new shock boundary conditions, whereas Paper II [C.P. Knisely and X. Zhong, “Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. II. Direct numerical simulation,” Phys. Fluids 31, 024104 (2019)] focuses on DNS with the overall goal of investigating the impact of supersonic modes on transition. LST results indicate that supersonic modes exist in the flow with wall-to-free-stream temperature ratio Tw/T∞ = 0.2 and create an abnormal growth pattern. However, supersonic modes were not shown to exist using LST in the case with Tw/T∞ = 0.667. Subsequent DNS analysis in Paper II shows supersonic modes in the Tw/T∞ = 0.667 case, although they are significantly weaker than the second mode and are unlikely to lead to transition. Understanding the mechanism of supersonic modes can yield more accurate transition location predictions leading to improved estimates for drag and heat transfer to the vehicle.

A conceptual model of the flame stabilization mechanisms for a lifted Diesel-type flame based on direct numerical simulation and experiments

This work presents an analysis of the stabilization of diffusion flames created by the injection of fuel into hot air, as found in Diesel engines. It is based on experimental observations and uses a dedicated Direct Numerical Simulation (DNS) approach to construct a numerical setup, which reproduces the ignition features obtained experimentally. The resulting DNS data are then used to classify and analyze the events that allow the flame to stabilize at a certain Lift-Off Length (LOL) from the fuel injector. Both DNS and experiments reveal that this stabilization is intermittent: flame elements first auto-ignite before being convected downstream until another sudden auto-ignition event occurs closer to the fuel injector. The flame topologies associated to such events are discussed in detail using the DNS results, and a conceptual model summarizing the observation made is proposed. Results show that the main flame stabilization mechanism is auto-ignition. However, multiple reaction zone topologies, such as triple flames, are also observed at the periphery of the fuel jet helping the flame to stabilize by filling high-temperature burnt gases reservoirs localized at the periphery, which trigger auto-ignitions.

Effect of non-zero relative velocity on the flame speed of two-phase laminar flames

A numerical study of one-dimensional n-heptane/air spray flames is presented. The objective is to evaluate the flame propagation speed in the case where droplets evaporate inside the reaction zone with possibly non-zero relative velocity. A Direct Numerical Simulation approach for the gaseous phase is coupled to a discrete particle Lagrangian formalism for the dispersed phase. A global two-step n-heptane/air chemical mechanism is used. The effects of initial droplet diameter, overall equivalence ratio, liquid loading and relative velocity between gaseous and liquid phases on the laminar spray flame speed and structure are studied. For lean premixed cases, it is found that the laminar flame speed decreases with increasing initial droplet diameter and relative velocity. On the contrary, rich premixed cases show a range of diameters for which the flame speed is enhanced compared to the corresponding purely gaseous flame. Finally, spray flames controlled by evaporation always have lower flame speeds. To highlight the controlling parameters of spray flame speed, approximate analytical expressions are proposed, which give the correct trends of the spray flame propagation speed behavior for both lean and rich mixtures.