Streamwise Diffusion Research Articles

Streamwise dispersion of soluble matter in solvent flowing through a tube

For the dispersion of soluble matter in solvent flowing through a tube as investigated originally by G.I. Taylor, a streamwise dispersion theory is developed from a Lagrangian perspective for the whole process with multi-scale effects. By means of a convected coordinate system to decouple convection from diffusion, a diffusion-type governing equation is presented to reflect superposable diffusion processes with a multi-scale time-dependent anisotropic diffusivity tensor. A short-time benchmark, complementing the existing Taylor–Aris solution, is obtained to reveal novel statistical and physical features of mean concentration for an initial phase with isotropic molecular diffusion. For long times, effective streamwise diffusion prevails asymptotically corresponding to the overall enhanced diffusion in Taylor's classical theory. By inverse integral expansions of local concentration moments, a general streamwise dispersion model is devised to match the short- and long-time asymptotic solutions. Analytical solutions are provided for most typical cases of point and area sources in a Poiseuille tube flow, predicting persistent long tails and skewed platforms. The theoretical findings are substantiated through Monte Carlo simulations, from the initial release to the Taylor dispersion regime. Asymmetries of concentration distribution in a circular tube are certified as originated from (a) initial non-uniformity, (b) unidirectional flow convection, and (c) non-penetration boundary effect. Peculiar peaks in the concentration cloud, enhanced streamwise dispersivity and asymmetric collective phenomena of concentration distributions are illustrated heuristically and characterised to depict the non-equilibrium dispersion. The streamwise perspective could advance our understanding of macro-transport processes of both passive solutes and active suspensions.

Nonlinear dynamics of a liquid film flow over a solid substrate in the presence of external shear stress and electric field

Using approximation techniques, long-wave length framework and boundary-layer, the effects of electrostatic force and induced shear stress on the flow behavior down an inclined solid substrate are investigated. In general case, the considered model accounts in the presence of inertia regime and streamwise viscous diffusion with the influence of normal electric field and an imposed shear stress. Using the Galerkin weighted residual, two coupled evolution equations for the flow rate and film thickness are extracted. In the appropriate limit cases, the evolution equations obtained by previous authors are recovered. The primary instability has been analyzed using the Whitham wave hierarchy framework. In the nonlinear regime, the behavior of solitary waves arose on the surface of liquid film due to the effects of electrostatic force and imposed shear stress throughout three-dimensional dynamical systems. Some bifurcation points are reported. In both extremely viscous and electrogravity regimes, the Benney-like equation is extracted in a new form. By excluding contribution of external shear stress and viscous dispersion parameter, the interesting results of previous authors are recovered. In both weakly nonlinear and inertialess regimes, the bifurcation points of the three dynamical systems are discussed within the Kuramoto–Sivashinsky type equation.

Probability Distributions of Particle Hop Distance and Travel Time Over Equilibrium Mobile Bedforms

The joint probability distribution of streamwise particle hop distance, lateral particle hop distance, and travel time constrains the relationships between topographic change and sediment transport at the granular scale. Previous studies have investigated the ensemble characteristics of particle motions over plane bed topography; however, it is unclear whether reported distributions remain valid when bedforms are present. Here, we present measurements of particle motion over bedform topography obtained in a laboratory flume and compare these to particle motions over plane bed topography with otherwise similar conditions. We find substantial differences in particle motion in the presence of bedforms that are relevant to macroscopic models of sediment transport. Most notably, bedforms increase the standard deviation of streamwise and lateral hop distances relative to the mean streamwise hop distance. This implies that bedforms increase the streamwise and lateral diffusion lengths and, equivalently, increase diffusive‐like fluxes.

Simulation of multi-component multi-phase fluid flow in two-dimensional anisotropic heterogeneous porous media using high-order control volume distributed methods

In this paper, flow of multi-component two-phase fluids in highly heterogeneous anisotropic two-dimensional porous media is studied using computational methods suitable for unstructured triangular and/or quadrilateral grids. The physical model accounts for miscibility and compressibility of fluids while gravity and capillary effects are neglected. The governing equations consist of a pressure equation together with a system of mass conservation equations. For solving pressure equation, a new method called Control Volume Distributed Finite Element Method (CVDFEM) is introduced which uses Control Volume Distributed (CVD) vertex-centered grids. It is shown that the proposed method is able to approximate the pressure field in highly anisotropic and heterogeneous porous media fairly accurately. Moreover the system of mass conservation equations is solved using various upwind and central schemes. These schemes are extended from one-dimensional to two-dimensional unstructured grids. Using a series of numerical test cases, comparison is made between different approaches for approximation of the hyperbolic flux function. Semi one-dimensional high-order data reconstruction procedures are employed to decrease stream-wise numerical diffusion. The results suggest that the Modified Dominant Wave (MDW) scheme outperforms other hyperbolic schemes studied in this paper from both accuracy and computational cost points of view.

Oscillatory Burner-Attached Diffusion Flame in a Viscous Vortex

ABSTRACTA method for mathematically solving the oscillatory infinite reaction rate diffusion flame is extended to the case where the oscillating convective coflow is either inside of, or adjacent to, a viscous vortex. The neglect of streamwise diffusion coupled with the restriction to infinite rate chemistry produces a Burke-Schumann boundary layer flame. The mathematical transformation, which does not require a priori restriction to small coflow oscillations, renders the transient oscillatory problem equivalent to a steady-state problem that can be solved mathematically and numerically evaluated to a high degree of accuracy. Flow fluctuations that are large fractions of the initial flow field are described exactly. Features of the flame response are examined without recourse to detailed time-dependent numerical simulations. There is no need to perform any small-perturbation analyses. The method is applied to examine the influences of the bulk inflow speed, the bulk inflow oscillation rate, and the interaction of the inflow and its oscillation rate with the viscous vortex.

How vegetation in flows modifies the turbulent mixing and spreading of jets

While studies on vegetated channel flows have been developed in many research centers, studies on jets interacting with vegetation are still rare. This study presents and analyzes turbulent jets issued into an obstructed cross-flow, with emergent vegetation simulated with a regular array of cylinders. The paper presents estimates of the turbulence diffusion coefficients and the main turbulence variables of jets issued into a vegetated channel flow. The experimental results are compared with jets issued into unobstructed cross-flow. In the presence of the cylinder array, the turbulence length-scales in the streamwise and transverse directions were reduced, relative to the unobstructed crossflow. This contributed to a reduction in streamwise turbulent diffusion, relative to the unobstructed conditions. In contrast, the transverse turbulent diffusion was enhanced, despite the reduction in length-scale, due to enhanced turbulent intensity and the transverse deflection of flow around individual cylinders. Importantly, in the obstructed condition, the streamwise and transverse turbulent diffusion coefficients are of the same order of magnitude.

Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles

This paper presents a novel subgrid scale (SGS) model for simulating convection-dominated species transport at deformable fluid interfaces. One possible application is the Direct Numerical Simulation (DNS) of mass transfer from rising bubbles. The transport of a dissolving gas along the bubble–liquid interface is determined by two transport phenomena: convection in streamwise direction and diffusion in interface normal direction. The convective transport for technical bubble sizes is several orders of magnitude higher, leading to a thin concentration boundary layer around the bubble. A true DNS, fully resolving hydrodynamic and mass transfer length scales results in infeasible computational costs. Our approach is therefore a DNS of the flow field combined with a SGS model to compute the mass transfer between bubble and liquid. An appropriate model-function is used to compute the numerical fluxes on all cell faces of an interface cell. This allows to predict the mass transfer correctly even if the concentration boundary layer is fully contained in a single cell layer around the interface. We show that the SGS-model reduces the resolution requirements at the interface by a factor of ten and more. The integral flux correction is also applicable to other thin boundary layer problems. Two flow regimes are investigated to validate the model. A semi-analytical solution for creeping flow is used to assess local and global mass transfer quantities. For higher Reynolds numbers ranging from Re=100 to Re=460 and Péclet numbers between Pe=104 and Pe=4⋅106 we compare the global Sherwood number against correlations from literature. In terms of accuracy, the predicted mass transfer never deviates more than 4% from the reference values.

Analysis of the Fluid Diffusion Characteristics in Platelet Pre-Mixer

Platelet mixer is a type of high-efficient equipment which can mix fluids through platelet micro-channels by cross-bedded injection. It overcomes disadvantages that the fluid is centralized around the inlet and it improves the diffusion. In this work, the platelet mixer is used for pre-mixer in a rocket engine, and the diffusion of oxygen and air in pre-mixer is analyzed. The high speed jet leads to a strong shear force near the inlet which produces counter-rotating stream-wise vortices around the jet. The angle between the direction of jet and radial direction influences both the stream-wise vortices and diffusion. The vortices intensity increases as the angle increase from 0o to 20o.

Theoretical and numerical analysis of oscillating diffusion flames

A novel method is presented for solving the forced transient diffusion flame in the exit region of a coflow burner. Streamwise diffusion is eliminated, which produces the Burke–Schumann model. A mathematical transformation renders the transient, forced convection problem equivalent to a steady-state convection problem. The transformation differs from previous approaches because its use does not require a priori restriction to small perturbations. For this reason, flow fluctuations that are large fractions of the initial flow field may be described exactly and features of nonlinear response can be examined without recourse to detailed numerical simulation. The method is applied to study flame evolution and oscillation for two physically separated coflow slot burner flames as they merge.

Film cooling effectiveness measurements of a near surface streamwise diffusion hole

This paper presents an experimental investigation on a new configuration of film cooling hole, namely the near surface streamwise diffusion (NSSD) hole. A discrete streamwise diffused shallow submerged structure is constructed on the exit of a simple cylindrical hole. There is a middle bulge at the exit edge of the shallow submerged structure to expand the coolant laterally. The adiabatic film cooling effectiveness was measured using pressure sensitive paint (PSP) technique on a flat plate model. The experiment was performed under density ratio DR=1.38 and the blowing ratio M=0.5–2.5. A simple cylindrical hole and a typical fan-shaped hole were chosen to be the baseline holes to be compared with NSSD hole. Six NSSD hole cases were tested and compared to investigate the influence of geometrical parameters. The experimental results showed that the cooling effectiveness of NSSD hole was significantly higher than the simple cylindrical hole but was slightly lower than the fan-shaped hole. The streamwise diffused shallow submerged structure can effectively expand the coolant laterally and can keep film attached to the surface under high blowing ratio. Due to the special flow mechanism, the laterally averaged cooling effectiveness of NSSD hole exhibited a ladder-style decreasing trend where the cooling effectiveness far downstream is close to the fan-shaped hole. The change of geometrical parameter of the near surface streamwise diffusion structure results in the obvious variation of cooling effectiveness especially at the high blowing ratio conditions. A smaller streamwise spacing between the middle bulge and the exit edge of the cylindrical hole can enhance the interaction between the coolant and the middle bulge, which is conducive to the increase of cooling effectiveness.

On the Kapitza instability and the generation of capillary waves

We revisit the classical problem of a liquid film falling along a vertical wall due to the action of gravity, i.e. the Kapitza paradigm (Kapitza,Zh. Eksp. Teor. Fiz., vol. 18, 1948, pp. 3–28). The free surface of such a flow is typically deformed into a train of solitary pulses that consists of large asymmetric wave humps preceded by small precursory ripples, designated as ‘capillary waves’. We set out to answer four fundamental questions. (i) By what mechanism do the precursory ripples form? (ii) How can they travel at the same celerity as the large-amplitude main humps? (iii) Why are they designated as ‘capillary waves’? (iv) What determines their wavelength and number and why do they attenuate in space? Asymptotic expansion as well as direct numerical simulations and calculations with a low-dimensional integral boundary-layer model have yielded the following conclusions. (i) Precursory ripples form due to an inertia-based mechanism at the foot of the leading front of the main humps, where the local free-surface curvature is large. (ii) The celerity of capillary waves is matched to that of the large humps due to the action of surface tension, which speeds up the former and slows down the latter. (iii) They are justly designated as ‘capillary waves’ because their wavelength is systematically shorter than the visco-capillary cutoff wavelength of the Kapitza instability. Due to a nonlinear effect, namely that their celerity decreases with decreasing amplitude, they nonetheless attain/maintain a finite amplitude because of being continuously compressed by the pursuing large humps. (iv) The number and degree of compression of capillary waves is governed by the amplitude of the main wave humps as well as the Kapitza number. Large-amplitude main humps travel fast and strongly compress the capillary waves in order for these to speed up sufficiently. Also, the more pronounced the first capillary wave becomes, the more (spatially attenuating) capillary waves are needed to allow a smooth transition to the back of the next main hump. These effects are amplified by decreasing the Kapitza number, whereby, at very small values, streamwise viscous diffusion increasingly attenuates the amplitude of the capillary waves.

Films in narrow tubes

AbstractWe consider the axisymmetric arrangement of an annular liquid film, coating the inner surface of a narrow cylindrical tube, in interaction with an active core fluid. We introduce a low-dimensional model based on the two-phase weighted residual integral boundary layer (WRIBL) formalism (Dietze & Ruyer-Quil, J. Fluid Mech., vol. 722, 2013, pp. 348–393) which is able to capture the long-wave instabilities characterizing such flows. Our model improves upon existing works by fully representing interfacial coupling and accounting for inertia as well as streamwise viscous diffusion in both phases. We apply this model to gravity-free liquid-film/core-fluid arrangements in narrow capillaries with specific attention to the dynamics leading to flooding, i.e. when the liquid film drains into large-amplitude collars that occlude the tube cross-section. We do this against the background of linear stability calculations and nonlinear two-phase direct numerical simulations (DNS). Due to the improvements of our model, we have found a number of novel/salient physical features of these flows. First, we show that it is essential to account for inertia and full interphase coupling to capture the temporal evolution of flooding for fluid combinations that are not dominated by viscosity, e.g. water/air and water/silicone oil. Second, we elucidate a viscous-blocking mechanism which drastically delays flooding in thin films that are too thick to form unduloids. This mechanism involves buckling of the residual film between two liquid collars, generating two very pronounced film troughs where viscous dissipation is drastically increased and growth effectively arrested. Only at very long times does breaking of symmetry in this region (due to small perturbations) initiate a sliding motion of the liquid film similar to observations by Lister et al. (J. Fluid Mech., vol. 552, 2006, pp. 311–343) in thin non-flooding films. This kickstarts the growth of liquid collars anew and ultimately leads to flooding. We show that streamwise viscous diffusion is essential to this mechanism. Low-frequency core-flow oscillations, such as occur in human pulmonary capillaries, are found to set off this sliding-induced flooding mechanism much earlier.

Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations

AbstractIn this paper we derive consistent shallow-water equations for the flow of thin films of power-law fluids down an incline. These models account for the streamwise diffusion of momentum, which is important to describe accurately the full dynamics of thin-film flows when instabilities such as roll waves arise. These models are validated through a comparison with the Orr–Sommerfeld equations for large-scale perturbations. We consider only laminar flow for which the boundary layer issued from the interaction of the flow with the bottom surface has an influence all over the transverse direction to the flow. In this case the concept itself of a thin film and its relation with long-wave asymptotics leads naturally to flow conditions around a uniform free-surface Poiseuille flow. The apparent viscosity diverges at the free surface, which, in turn, introduces a singularity in the formulation of the Orr–Sommerfeld equations and in the derivation of shallow-water models. We remove this singularity by introducing a weaker formulation of the Cauchy momentum equations. No regularization procedure is needed, nor any distinction between shear thinning and thickening cases. Our analysis, though, is only valid when the flow behaviour index $n$ is larger than $1/ 2$, and strongly suggests that the Cauchy momentum equations are ill-posed if $n\leq 1/ 2$.

The emergence of localized vortex–wave interaction states in plane Couette flow

AbstractThe recently understood relationship between high-Reynolds-number vortex–wave interaction theory and computationally generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex–wave interaction (VWI) theory is used in the long streamwise wavelength limit to continue the development found at order-one wavelengths by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). The asymptotic description given reduces the Navier–Stokes equations to the so-called boundary-region equations, for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from $O(h)$ to $O(Rh)$, where $R$ is the Reynolds number and $2h$ is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave–vortex states. The solutions we calculate have an asymptotic error proportional to ${R}^{- 2} $ when compared to the full Navier–Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel, so that the roll part of the flow is driven throughout the flow rather than as in Hall & Sherwin as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling, which, when approached, causes the flow to localize and take on similarities with computationally generated or experimentally observed turbulent spots. In effect, the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier–Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

Wavy regime of a power-law film flow

AbstractWe consider a power-law fluid flowing down an inclined plane under the action of gravity. The divergence of the viscosity at zero strain rate is taken care of by introducing a Newtonian plateau at small strain rate. Two-equation models are formulated within the framework of lubrication theory in terms of the exact mass balance and an averaged momentum equation, which form a set of evolution equations for the film thickness$h$, a local velocity amplitude or the flow rate$q$. The models account for the streamwise diffusion of momentum. Comparisons with Orr–Sommerfeld stability analysis and with direct numerical simulation (DNS) show convincing agreement in both linear and nonlinear regimes. The influence of shear-thinning or shear-thickening on the primary instability is shown to be non-trivial. A destabilization of the base flow close to threshold is promoted by the shear-thinning effect, whereas, further from threshold, it tends to stabilize the base flow when the viscous damping of short waves becomes dominant. A reverse situation is observed in the case of shear-thickening fluids. Shear-thinning accelerates solitary waves and promotes a subcritical onset of travelling waves at larger wavenumber than the linear cut-off wavenumber. A conditional stability of the base flow is thus observed. This phenomenon results from a reduction of the effective viscosity at the free surface. When compared with DNS, simulations of the temporal response of the film based on weighted residual models satisfactorily capture the conditional stability of the film.

Analytical puff modelling of light-wind dispersion in stable and unstable conditions

The puff modelling technique can describe dispersion in light winds by accounting for streamwise diffusion. A simple analytical puff model is formulated for stable as well as unstable stratification. In the stable case, the model is a slight generalisation of the Gaussian puff model of Thomson and Manning (2001), whereas in the unstable case it incorporates a probability density function approach to account for the observed skewness of the vertical turbulent velocities. An evaluation conducted using a well-known dataset collected under stable conditions suggests that the performance of the model is comparable to some of the top-performing models based on more complex techniques. However, it is apparent that some of the observed complex dispersion behaviour that arises due to the non-Gaussian characteristics of the wind direction distribution cannot be properly modelled by using the hourly-averaged meteorology. Since there are no suitable datasets for evaluation under unstable conditions, the puff model is benchmarked against a full three-dimensional Lagrangian particle model. The results show that the puff model is capable of describing most of the well-known features of convective dispersion observed under moderate winds, and that it simulates light-wind dispersion characteristics that are in good agreement with the particle model predictions. The results also highlight that the time since the start of tracer release is an important parameter in light winds. Quantitative estimates as to when the effects of streamwise diffusion in unstable flows can be neglected are also given.

Characteristics of vertically injected buoyant jets of highly diluted propane

In coflow jets with the nozzle diameter of O (1 cm) and the fuel jet velocity of O (10 cm/s), the buoyancy induced by the density difference between the fuel and air influences the jet structure appreciably. The present study investigated the behavior of such a buoyant jet numerically and experimentally, especially when the fuel stream had higher density than air. When the fuel jet was composed of propane highly diluted with nitrogen, the fuel jet was decelerated and formed a stagnation region. Consequently, the fuel was carried downstream by the coflow having a circular cone shape. When the fuel was moderately diluted or as the jet velocity increased, numerical results showed the Kelvin-Helmholtz type instability along the mixing layer of the jet. When the fuel jet velocity was relatively high, the stagnation height increased nonlinearly with fuel jet velocity having the power of approximately 1.62. In the relatively high Reynolds number regime of Re > 80, the stagnation height can be correlated to Re0.62Ri−0.5, indicating the combined effects of buoyancy and jet momentum. As the Reynolds number becomes small, the stagnation height was affected by the streamwise diffusion due to fuel concentration gradient and by the wake behavior near the nozzle tip. Accordingly, the stagnation heights approach to none-zero values, which were found to be relatively insensitive to fuel dilution.

Applying Taguchi’s off-line quality control method and ANOVA on the maneuverability of the F-5E intake

Air quality leading up to the compressor face of a fighter aircraft determines the engine performance considerably. A deficiency in the quality could lead to flutter or stall in the engines. In this study, two statistical methods; the Taguchi Method (TM) and the Analysis of Variance (ANOVA) are used to evaluate airflow quality through the intake via fighter aircraft maneuvers. The three factors associated directly with aircraft maneuverability are the Mach number (M), Angles of Incident ( α ) and Sideslip ( β ). Desirable air quality can be described as having high pressure recoveries as well as low distortion at the Aerodynamic Interface Plane (AIP). The intake studied is the port side F-5E duct. Results show that an increase in the Mach number affects the streamwise diffusion of the fluid more than the changing the angles of attack and sideslip, resulting in lower pressure recovery. The secondary flow formation in the streamwise direction is unable to dissipate and increases in strength with increasing Mach number. The curvature in the z -axis is more pronounced than that existing in the x -axis, leading to the formation of more adverse pressure gradients forming and hence greater secondary flow strength. This results in a more distorted flow leading to the AIP. This observation is in tandem with the values of the DC (60) readings obtained. The F-5E’s Taguchi’s Method results show that Mach number had the greatest effect on pressure recovery, and AOA affected distortion most considerably. Results from ANOVA show that Factors A, B and C and Interactions AC and BC affect the distortion of airflow. However, Factor B or the angle of attack affects this distortion most significantly.

Film flows down a fiber: Modeling and influence of streamwise viscous diffusion

A two-equation model is formulated in terms of two coupled evolution equations for the film thickness h and the local flow rate q within the framework of lubrication theory. Consistency is achieved up to first order in the film parameter epsilon and streamwise diffusion effects are accounted for. The evolution equation obtained by Craster and Matar [1] is recovered in the appropriate limit. Comparisons to the experimental results by [2] and [3] show good agreement in the linear and nonlinear regimes. Second-order viscous diffusion terms are found to potentially enhance the speed and amplitude of nonlinear waves triggered by the Rayleigh-Plateau instability mechanism. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise diffusion on the dynamics of the flow and the wave selection process.

Infrared imaging of AP/HTPB laminate propellant flames

The combustion of ammonium perchlorate (AP), hydroxyl-terminated polybutadiene (HTPB) two-dimensional laminate propellants is investigated using simultaneous infrared (IR) and ultraviolet (UV) emission imaging at 1–50 atm. The IR (3300–3900 nm) measurements sense flame presence nominally via HCl rovibrational excitation and the UV (310 nm) measurements via OH electronic excitation, although continuum emission is also present. The following findings are made. (1) The new IR results generally confirm the flame-structure regime map (split vs merged flames) obtained previously from UV observations for both pure and oxygenated fuel-binders. Here “merged” refers to lateral merging of the two diffusion flame branches (one for each fuel–AP interface) as pressure and fuel-layer thickness are reduced. (2) A new finding (not seen previously in the UV) is that for pure binder systems at relatively low pressures ( < 15 atm ) , IR imaging clearly shows two stream-wise distinct diffusion flames, which are possibly the primary diffusion flame (PDF) and final diffusion flame (FDF) structures postulated in the BDP model. (3) As pressure increases the PDF and FDF flames observed for pure binder merge stream-wise and become indistinguishable. (4) Oxygenated fuel-binder (75:25 fine-AP/HTPB-binder) laminates do not show distinct PDF and FDF flames. (5) The IR imaging technique improves on the UV in terms of surface definition because of the differences in IR versus UV absorption/emission properties of fuel-rich gaseous species near the burning surface: oxygenated binder surfaces that appear protruding in UV in the split-flame regime are less so in IR. (6) Computational results using just the AP-premixed and final diffusion flames capture the flame-surface structure regime map qualitatively but not quantitatively, further underscoring the importance of the primary diffusion flame.