Profiles Of Dissipation Rate Research Articles

Energetic Stratified Turbulence Generated by Kuroshio–Seamount Interactions in Tokara Strait

Abstract Generating mechanisms and parameterizations for enhanced turbulence in the wake of a seamount in the path of the Kuroshio are investigated. Full-depth profiles of finescale temperature, salinity, horizontal velocity, and microscale thermal-variance dissipation rate up- and downstream of the ∼10-km-wide seamount were measured with EM-APEX profiling floats and ADCP moorings. Energetic turbulent kinetic energy dissipation rates and diapycnal diffusivities above the seamount flanks extend at least 20 km downstream. This extended turbulent wake length is inconsistent with isotropic turbulence, which is expected to decay in less than 100 m based on turbulence decay time of N−1 ∼ 100 s and the 0.5 m s−1 Kuroshio flow speed. Thus, the turbulent wake must be maintained by continuous replenishment which might arise from (i) nonlinear instability of a marginally unstable vortex wake, (ii) anisotropic stratified turbulence with expected downstream decay scales of 10–100 km, and/or (iii) lee-wave critical-layer trapping at the base of the Kuroshio. Three turbulence parameterizations operating on different scales, (i) finescale, (ii) large-eddy, and (iii) reduced-shear, are tested. Average ε vertical profiles are well reproduced by all three parameterizations. Vertical wavenumber spectra for shear and strain are saturated over 10–100 m vertical wavelengths comparable to water depth with spectral levels independent of ε and spectral slopes of −1, indicating that the wake flows are strongly nonlinear. In contrast, vertical divergence spectral levels increase with ε.

A generalized mixture fraction formulation for nonpremixed ammonia–hydrogen flames

For nonpremixed combustion, the mixture fraction variables are widely used as a conserved scalar to map out the flame structure in a unified manner. For ammonia–hydrogen flames, however, a proper definition of the mixture fraction variable is difficult due to the presence of fuel-bound nitrogen in ammonia. This work proposes a revised mathematical definition of the mixture fraction as a modified form of the Bilger’s formula in order to describe the characteristics of nonpremixed ammonia–hydrogen blend flames at general fuel mixture ratios. The newly defined mixture fraction properly accounts for differential diffusion effects in the presence of hydrogen, and thus serves as a convenient invariant coordinate for the flame manifolds. Counterflow nonpremixed flames with different ammonia–hydrogen ratios were computed and the validity of the new formulation is demonstrated. The analysis of the flame structure revealed the presence of double reaction zones, one dominated by the hydrogen oxidation and the other by ammonia oxidation. The role of differential diffusion is crucial in determining the existence of double reactive layers and its impact is evaluated by varying the strain rate. Detailed reaction analysis revealed that the hydrogen flame is related to HO2 reactions, while the ammonia flame to N, NH2 and NH3 reaction groups. As the strain rate is increased, OH reactions are responsible for shifting dominant heat release rate from the hydrogen flame to the ammonia flames. The scalar dissipation rate profile shows two distinct peaks, suggesting that a different functional description is needed in the flamelet modeling framework.

A rapid method for flow pattern, mixing time estimation and turbulent dissipation rates in turbulent stirred mixers based on 2-D network-of-zones (NoZ) model

A rapid non-iterative method for estimating flow patterns in stirred vessels in fully turbulent regime was defined and evaluated starting from limited known data near the core regions based on the 2-D network-of-zone (NoZ) model. Flow pattern induced by impeller of arbitrary diameter could be generated non-iteratively from known NoZ data with impeller of same type but another diameter, before mixing time was estimated within minutes on a standard PC. Case studies on six different impeller diameter and types representing typical radial flow impellers and axial flow impellers respectively was presented, errors of mixing time predicted in which were all within a maximum of ±33% compared to mixing time predicted via experimental correlation. A rapid method for estimating turbulent dissipation rate profiles were also introduced and evaluated quantitatively.

Estimating Profiles of Dissipation Rate in the Upper Ocean Using Acoustic Doppler Measurements Made from Surface-Following Platforms

Abstract High-resolution profiles of vertical velocity obtained from two different surface-following autonomous platforms, Surface Wave Instrument Floats with Tracking (SWIFTs) and a Liquid Robotics SV3 Wave Glider, are used to compute dissipation rate profiles ϵ(z) between 0.5 and 5 m depth via the structure function method. The main contribution of this work is to update previous SWIFT methods to account for bias due to surface gravity waves, which are ubiquitous in the near-surface region. We present a technique where the data are prefiltered by removing profiles of wave orbital velocities obtained via empirical orthogonal function (EOF) analysis of the data prior to computing the structure function. Our analysis builds on previous work to remove wave bias in which analytic modifications are made to the structure function model. However, we find the analytic approach less able to resolve the strong vertical gradients in ϵ(z) near the surface. The strength of the EOF filtering technique is that it does not require any assumptions about the structure of nonturbulent shear, and does not add any additional degrees of freedom in the least squares fit to the model of the structure function. In comparison to the analytic method, ϵ(z) estimates obtained via empirical filtering have substantially reduced noise and a clearer dependence on near-surface wind speed.

Data-based instantaneous conditional progress variable dissipation rate modeling for turbulent premixed combustion

Manifold-based turbulent combustion modeling reduces computational cost considerably by projecting the thermochemical state onto a lower-dimensional space of a few manifold variables and reconstructing the state from a set of manifold equations. In turbulent premixed combustion, a one-dimensional space is defined by the progress variable, and the progress variable dissipation rate appears as a parameter in the resulting one-dimensional manifold equations. Closure of the manifold equations requires a model for the dependence of this dissipation rate on the progress variable. Existing models for these profiles are often ad hoc, derived analytically with crude assumptions, and assumed to be spatially and temporally ‘universal’ — yet the resulting thermochemical state can be sensitive to these profiles. To evaluate the assumptions in traditional models, instantaneous conditional progress variable dissipation rate profiles are extracted from DNS databases of turbulent premixed jet flames. At low Karlovitz number, the instantaneous conditional progress variable dissipation rate profiles exhibit some variance but are relatively well-characterized by a ‘universal’ model. However, a large variance in the instantaneous conditional progress variable dissipation rate profiles is observed at high Karlovitz number, indicating the form of the conditional dissipation rate is certainly not ‘universal’. To capture this variation at high Karlovitz number, a data-based model for the instantaneous conditional progress variable dissipation rate profiles is generated, and the resulting model predictions demonstrate excellent agreement with profiles extracted from DNS and far less error than ‘universal’ models. The sensitivity of the solutions of the one-dimensional premixed manifold equations to the dissipation rate profile is demonstrated a priori, and the error associated with the observed variance in the instantaneous conditional progress variable dissipation rate profiles (i.e., relative to the DNS mean) indicates that an instantaneous modeling approach is necessary. A variable importance study is conducted, and the conditional velocity gradient is identified as the critical quantity that characterizes the observed variance in the instantaneous conditional progress variable dissipation rate profiles, providing further motivation for a data-based modeling approach.

Investigation of Turbulent Dissipation Rate Profiles from Two Radar Wind Profilers at Plateau and Plain Stations in the North China Plain

Turbulence is ubiquitous in the planetary boundary layer (PBL), which is of great importance to the prediction of weather and air quality. Nevertheless, the profiles of turbulence in the whole PBL as observed by radar wind profilers (RWPs) are rarely reported. In this communication, the purpose was to investigate the vertical structures of turbulence dissipation rate (ε) obtained from the Doppler spectrum width measurements from two RWPs at plateau (Zhangbei) and plain (Baoding) stations in the North China Plain for the year 2021, and to tease out the underlying mechanism for the difference of ε between Zhangbei and Baoding. Under clear-sky conditions, the annual mean value of ε in the PBL over the plateau station was found to be higher than that over the plain station throughout the daytime from 0900 to 1700 local standard time. The magnitude of ε at both stations showed significant seasonal variation, with the strongest ε in summer but the weakest in winter. If a larger difference between the 2 m air temperature and surface temperature (Ta−Ts), as a surrogate of sensible heat flux, is observed, the turbulence intensity tends to become stronger. The influence of vertical wind shear on turbulence was also analyzed. Comparison analyses showed that the plateau station of Zhangbei was characterized by larger sensible heat flux and stronger wind shear compared with the plain station of Baoding. This may account for the more intense ε within the PBL of Zhangbei. Moreover, the magnitude of ε in the PBL was positively correlated with the values of both Ta−Ts and vertical wind shear. The findings highlight the urgent need to characterize the vertical turbulence structure in the PBL over a variety of surfaces in China.

Boundary Layer Energetics of Rapid Wind and Wave Forced Mixing Events

Abstract The observed development of deep mixed layers and the dependence of intense, deep-mixing events on wind and wave conditions are studied using an ocean LES model with and without an imposed Stokes-drift wave forcing. Model results are compared to glider measurements of the ocean vertical temperature, salinity, and turbulence kinetic energy (TKE) dissipation rate structure collected in the Icelandic Basin. Observed wind stress reached 0.8 N m−2 with significant wave height of 4–6 m, while boundary layer depths reached 180 m. We find that wave forcing, via the commonly used Stokes drift vortex force parameterization, is crucial for accurate prediction of boundary layer depth as characterized by measured and predicted TKE dissipation rate profiles. Analysis of the boundary layer kinetic energy (KE) budget using a modified total Lagrangian-mean energy equation, derived for the wave-averaged Boussinesq equations by requiring that the rotational inertial terms vanish identically as in the standard energy budget without Stokes forcing, suggests that wind work should be calculated using both the surface current and surface Stokes drift. A large percentage of total wind energy is transferred to model TKE via regular and Stokes drift shear production and dissipated. However, resonance by clockwise rotation of the winds can greatly enhance the generation of inertial current mean KE (MKE). Without resonance, TKE production is about 5 times greater than MKE generation, whereas with resonance this ratio decreases to roughly 2. The results have implications for the problem of estimating the global kinetic energy budget of the ocean.

Turbulence near a sandbar island in the lower Missouri River

AbstractRiver turbulence is spatially variable due to interactions between morphology of rivers and physical mechanics of flowing water. Understanding the variation of turbulence in rivers is important for characterizing transport processes of soluble and particulate materials in these systems. We present an exploratory effort to understand ecologically relevant flow patterns using measurements of mean flow and turbulence in a highly engineered river channel around an island in the lower Missouri River. Specifically, the profiles of mean river velocities were investigated to examine the logarithmic relation and associated parameters, including shear velocity and bed roughness. Turbulence intensity and Reynolds shear stress were compared with classic open‐channel profiles and previously reported river data in the hydraulics literature. With the capability of pulse‐to‐pulse coherent Doppler velocity profiling in high spatial resolution, we estimated the profiles of turbulence dissipation rate using resolved one‐dimensional velocity spectra. These measurement data allow us to examine the validity of turbulence production‐dissipation balance and the classic open‐channel profiles of turbulence statistics, including turbulence intensity, Reynolds shear stress, dissipation rate, and eddy viscosity. The field data show a strong variation of turbulence profiles in close vicinity of the river island. In shallow water depths close to the island, turbulence is substantially enhanced in comparison with classic open‐channel profiles. Such turbulence enhancement is likely attributed to non‐uniformity of the flow structures.

A general methodology for extracting and categorising flow patterns in turbulent stirred tank mixers based on 2-D network-of-zones (NoZ) model

A general methodology for characterising stirred tank mixers using two-dimensional network-of-zones (2-D NoZ) model was introduced in this work. Starting from high resolution 3-D velocity field database such as CFD simulations, flow patterns were preserved and simplified into 2-D inter-zonal flowrates and mass imbalances were corrected in the defined routine. Mixing time estimation and turbulent dissipation rate profiles were chosen as metrics for evaluation of the generated 2-D NoZ model, which were compared to CFD passive scalar and literature results on two typical stirred tank geometries covering both radial flow and axial flow impellers.

Winter Vertical Diffusion Rates in the Arctic Ocean, Estimated From 7Be Measurements and Dissipation Rate Profiles

AbstractOcean turbulent mixing is a key process affecting the uptake and redistribution of heat, carbon, nutrients, oxygen and other dissolved gasses. Vertical turbulent diffusivity sets the rates of water mass transformations and ocean mixing, and is intrinsically an average quantity over process time scales. Estimates based on microstructure profiling, however, are typically obtained as averages over individual profiles. How representative such averaged diffusivities are, remains unexplored in the quiescent Arctic Ocean. Here, we compare upper ocean vertical diffusivities in winter, derived from the 7Be tracer‐based approach to those estimated from direct turbulence measurements during the year‐long Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) expedition, 2019–2020. We found that diffusivity estimates from both methods agree within their respective measurement uncertainties. Diffusivity estimates obtained from dissipation rate profiles are sensitive to the averaging method applied, and the processing and analysis of similar data sets must take this sensitivity into account. Our findings indicate low characteristic diffusivities around 10−6 m2 s−1 and correspondingly low vertical heat fluxes.

Similarity of the turbulent kinetic energy dissipation rate distribution in the upper mixed layer of the tropical Indian Ocean

Turbulence within the upper ocean mixed layer plays a key role in various physical, biological, and chemical processes. Between September and November 2011, a dataset of 570 vertical profiles of the turbulent kinetic energy (TKE) dissipation rate, as well as conventional hydrological and meteorological data, were collected in the upper layer of the tropical Indian Ocean. These data were used to statistically analyze the vertical distribution of the TKE dissipation rate in the mixed layer. The arithmetic-mean method made the statistical TKE dissipation rate profile more scattered than the median and geometric-mean methods. The statistical TKE dissipation rate were respectively scaled by the surface buoyancy flux and the TKE dissipation rate at the mixed-layer base. It was found that the TKE dissipation rate scaled by that at the mixed-layer base exhibited better similarity characteristics than that scaled by the surface buoyancy flux, whether the stability parameter D/|LMO| was greater or less than 10, indicating that the TKE dissipation rate at the mixed-layer base is a better characteristic scaling parameter for reflecting the intrinsic structure of the TKE dissipation rate in the mixed layer, where D and LMO are respectively the mixed-layer thickness and the Monin-Obukhov length scale. The parameterization of the TKE dissipation rate at the mixed-layer base on the shear-driven dissipation rate and the surface buoyancy flux was further explored. It was found that the TKE dissipation rate at the mixed-layer base could be well fitted by a linear combination of three terms: the wind-shear-driven dissipation rate, the surface buoyancy flux, and a simple nonlinear coupling term of these two .

Upper‐Ocean Turbulence Structure and Ocean‐Ice Drag Coefficient Estimates Using an Ascending Microstructure Profiler During the MOSAiC Drift

AbstractSea ice mediates the transfer of momentum, heat, and gas between the atmosphere and the ocean. However, the under‐ice boundary layer is not sufficiently constrained by observations. During the Multidisciplinary drifting Observatory for the Study of the Arctic Climate (MOSAiC), we collected profiles in the upper 50–80 m using a new ascending vertical microstructure profiler, resolving the turbulent structure within 1 m to the ice. We analyzed 167 dissipation rate profiles collected between February and mid‐September 2020, from 89°N to 79°30′N through the Amundsen Basin, Nansen Basin, Yermak Plateau, and Fram Strait. Measurements covered a broad range of forcing (0–15 m s−1 wind and 0–0.4 m s−1 drift speeds) and sea ice conditions (pack ice, thin ice, and leads). Dissipation rates varied by over 4 orders of magnitude from 10−9 W kg−1 below 40 m to above 10−5 W kg−1 at 1 m. Following wind events, layers with dissipation W kg−1 extended down to 20 m depth under pack ice. In leads in the central Arctic, turbulence was enhanced 2–10 times relative to thin ice profiles. Under‐ice dissipation profiles allowed us to estimate the boundary layer thickness (4 ± 2 m), and the friction velocity (1–15 mm s−1, 4.7 mm s−1 on average). A representative range of drag coefficient for the MOSAiC sampling site was estimated to (4–6) × 10−3, which is a typical value for Arctic floe observations. The average ratio of drift speed to wind speed was close to the free‐drift ratio of 2% with no clear seasonal or regional variability.

A full year of turbulence measurements from a drift campaign in the Arctic Ocean 2019–2020

Ocean turbulent mixing is a key process in the global climate system, regulating ocean circulation and the uptake and redistribution of heat, carbon, nutrients, oxygen and other tracers. In polar oceans, turbulent heat transport additionally affects the sea ice mass balance. Due to the inaccessibility of polar regions, direct observations of turbulent mixing are sparse in the Arctic Ocean. During the year-long drift expedition “Multidisciplinary drifting Observatory for the Study of Arctic Climate” (MOSAiC) from September 2019 to September 2020, we obtained an unprecedented data set of vertical profiles of turbulent dissipation rate and water column properties, including oxygen concentration and fluorescence. Nearly 1,700 profiles, covering the upper ocean down to approximately 400 m, were collected in sets of 3 or more consecutive profiles every day, and complemented with several intensive sampling periods. This data set allows for the systematic assessment of upper ocean mixing in the Arctic, and the quantification of turbulent heat and nutrient fluxes, and can help to better constrain turbulence parameterizations in ocean circulation models.

Mixing in the Philippine Sea: Geography variability and parameterization

The Philippine Sea Mixing Experiment (PhiMix) was carried out in 2021 concurrently from two research vessels, obtaining microstructure measurements and fine structure-based estimates of turbulence and diffusivity covering nearly the whole Philippine basin for the first time. The inhomogeneous spatial distribution and weak magnitudes of turbulent mixing in the Philippine Sea basin were revealed, according to the upper-ocean turbulence measurements and full-depth CTD datasets. The averaged diffusivities in the Philippine Basin were ∼10−2 m2s−1 in the surface mixed layer and ∼10−5.5 m2s−1 in the ocean interior. Below the thermocline, the diffusivities generally increased and reached 10−4 m2s−1 above rough topography. Centered on the Palau Ridge, a significant east-west asymmetry was noted, with stronger mixing on the east side of the ridge. Additionally, the diapycnal diffusivity's spatial variability coincided with that inferred from tidal mixing between 500 and 3000 m. Typical dissipation rate and diapycnal diffusivity profiles in the Philippine Sea differed from the global averaged profiles, which form the basis of most existing mixing parameterizations. Here we proposed an improved parameterization scheme based on these new direct microstructure observations in the Philippine Sea. The new mixing parameterization scheme (PhiMixPar-PMP) performed well for the Northwest Pacific Ocean in ocean simulations. This regional study on mixing properties in the Philippine Sea can fill a knowledge gap in the big picture of global mixing geography and improve turbulence parameterizations in the ocean models.

Ocean Mixed Layer Depth From Dissipation

AbstractThe mixed layer depth (MLD) is a widely used parameter for physical, chemical, and biological oceanography. The MLD delimits that region of the ocean, which is directly influenced by the atmosphere. There is a similar length scale referred to as the mixing layer depth, which is determined from profiles of dissipation rate of turbulent kinetic energy, but is less prevalent due to the requirement for specialized instrumentation. Here, we suggest that the MLD can be estimated using density/temperature (henceforth hD) or using dissipation (henceforth hϵ). Utilizing two data sets in the North Atlantic collected with the autonomous Air‐Sea Interaction Profiler, hD and hϵ in the upper 100 m are compared. A new method based on the shape of the dissipation profile for estimating hϵ is presented. The main sources of turbulence in the upper ocean (i.e., wind, waves, and buoyancy fluxes), which tend to deepen the MLD, are more strongly correlated with hϵ variability for both data sets, confirming that the MLD derived from dissipation measurements represents more accurately the variability in the mixed layer. Given that dissipation measurements have become more operational due to the development of ocean microstructure technology, then it is appropriate that hϵ be adopted in the future for estimation of the MLD. This is additionally supported by the fact that hD typically overestimates the MLD when compared to hϵ, thus having consequences for studies where hD is used to represent the extent of mixing in the upper ocean.

Steady and unsteady turbulent flows and pollutant dispersion in 2D street canyons with novel boundary conditions and various Re numbers

The unsteady approaching flow (UAF) and its influence upon the flow pattern within street canyon are investigated from a perspective of mechanism in this study. Based on wind tunnel test (WTT) measurements, the boundary conditions for the approaching flow over 2D street canyons (H/W = 2.4) are described in continuous functions of reference velocity and height. The turbulence kinetic energy (TKE) profiles present a normal-log distribution in WTT; the turbulence eddy dissipation rate (TED) profiles are modified according to WTTs and basic governing equations. With a sinusoidally varying reference velocity, these novel boundary conditions are then applied to study the influence of UAF (with periodic sinusoidal fluctuations) upon the flow pattern within 2D street canyons. CFD simulations with proposed scenario agree well with WTT for steady and unsteady approaching flows regarding velocity magnitude, vortex position and TKE. UAF can also enhance street canyon's ventilation as pollutant decays faster than that under steady approaching flow. The proposed scenario is finally extended to the investigation of Re independence and shows the larger normalized TKE at the periphery prolongs the procedure of reaching Re independence. Once it reaches Re independence, the flow pattern under a larger TKE has a higher eddy centre and velocity magnitude.

Experimental and modeling study on meso- and micromixing in the rotor–stator spinning disk reactor

The micromixing efficiency in the rotor–stator spinning disk reactor is studied for increasing injection flow rates of limiting reagent. Using both the Villermaux–Dushman method and second Bourne reaction, an increase in segregation index was found with increasing injection flow rate. Mechanistic models describing micro- and mesomixing were reviewed and implemented to describe the effects of increasing segregation index with increasing injection flow rate. A simplified turbulent-dispersion model was found to describe this trend effectively. A new parameter, based on a logistic function, was proposed to couple mesomixing by turbulent-dispersion and micromixing by engulfment on basis of the ratio of the characteristic times for turbulent-dispersion and engulfment. For injection flow rates where micromixing was calculated to be dominant, the engulfment model was implemented. An experimental relation was fitted to describe the dependency of the local energy dissipation rate on the turbulent diffusion coefficient in the rotor–stator spinning disk reactor. For the Villermaux–Dushman test reaction and the second Bourne reaction, discrepancies were found in the local energy dissipation rates experienced by the reaction zone. These discrepancies were analyzed on basis of the reaction times of both test reactions. The effect of the reaction time was accounted for by introducing a time-dependent local energy dissipation rate, which could be used to study the energy dissipation rate profile in the rotor–stator spinning disk reactor.

Turbulent flow mapping in a high-flow tidal channel using mobile acoustic Doppler current profilers

In this investigation, instrumented mobile platforms are used to spatially map the turbulent flows in Grand Passage, one of the Bay of Fundy's more energetic tidal channels in Nova Scotia, Canada. The aim is to characterize the flow around the PLAT-I floating tidal energy platform developed by Sustainable Marine Energy Canada (SMEC). GPS-tracked surface drifters equipped with fast-sampling acoustic Doppler current profilers (ADCPs) provide turbulence-resolving vertical profiles of velocity and turbulent kinetic energy dissipation rate along drifter trajectories, while vessel-mounted ADCP transects complement the mean flow velocity measurements. These data are used to construct tridimensional quasi-synoptic maps of mean velocities and turbulence parameters for several stages of the tide around PLAT-I's location including peak ebb and flood currents. The data set includes measurements under natural flow conditions, and while the turbines installed on PLAT-I were both not operational and operational. The measurement techniques and resulting maps successfully capture the spatial and temporal structure of the flow for unsteady conditions for various tidal conditions. The combined wake of the four 6.3 m operational PLAT-I turbines was measured for a single tidal stage. Closer to the turbines, the vertical extent of the wake is about 6 m (approximately one turbine diameter), increasing with distance downstream, while the maximum observed velocity deficit is 26% relative to not operating conditions, decreasing with distance downstream.

Optimization of liquid jet ejector geometry and its impact on flow fields

Liquid jet liquid (LJL) ejectors have wide industrial application and the improvement in their efficiency is sought to increase the viability of their use. This study performs the optimization of the entire geometry of an LJL ejector with multiple parameters to maximize energy efficiency. The approach with multiple parameters allows to identify key geometrical features and interdependent parameters, like nozzle position and mixing chamber length, with respect to performance. Computational fluid dynamics (CFD) and optimization simulations were performed to optimize the parameters of Bézier curves that characterize the device's geometry. The optimization results showed that the ejector efficiency curve is sensitive to the nozzle and the suction chamber geometries. Simulations that consider the nozzle diameter and the nozzle position along ejector axis (NXP) as parameters of the optimization process resulted in higher efficiency values than those that kept these parameters fixed. The optimization of the diffuser curve, including the diameter of the diffuser and the length of the mixing chamber, also contributed to increase the efficiency of the device. It was observed that the increase in the length of the mixing chamber and the spacing of the nozzle imply similar effects in the efficiency and in the pressure, velocity and energy dissipation rate profiles confirming some correlation of these parameters on the performance. One may observe the effect of geometry modification through optimization on the flow profiles in key sections of the ejector: the flow profiles in the optimized geometry tend to be more homogenous, hence less dissipative, and it is also confirmed by local energy dissipation rate.

Two-dimensional manifold equations for multi-modal turbulent combustion: Nonpremixed combustion limit and scalar dissipation rates

Manifold-based combustion models have traditionally been limited to asymptotically nonpremixed or premixed combustion relying on one-dimensional manifold equations in terms of either mixture fraction or progress variable, respectively. More recently, multiple formulations of two-dimensional manifold equations have been proposed for multi-modal combustion in terms of both mixture fraction and some progress variable. However, none of these formulations simultaneously satisfies three desirable properties: computationally solvable manifold equations, an explicit transport equation for the progress variable, and the recovery of (quasi-)one-dimensional manifold equations for all asymptotic modes of combustion, a notable challenge for asymptotically nonpremixed combustion that requires a spatially homogeneous progress variable. In this work, a formulation for two-dimensional manifold equations in mixture fraction and generalized progress variable is developed that simultaneously satisfies all three of these desirable properties. Utilizing its flexibility, the generalized progress variable is constructed to be exactly spatially homogeneous for asymptotically nonpremixed combustion by explicitly relating its definition to solutions of one-dimensional manifold equations in mixture fraction. This definition of the generalized progress variable leads to trivial simplification of the two-dimensional manifold equations into (quasi-)one-dimensional equations in mixture fraction for asymptotically nonpremixed combustion. Within these two-dimensional manifold equations, three scalar dissipation rates (the mixture fraction dissipation rate, the generalized progress variable dissipation rate, and the cross-dissipation rate) appear as coefficients, and, to close the manifold equations, models are required for the dependence of these dissipation rates on the mixture fraction and generalized progress variable. Analytical model forms for the scalar dissipation rates, the development of which is aided by the new definition for the generalized progress variable, are compared against simulations of a laminar lifted coflow flame, and existing models for the mixture fraction and generalized progress variable dissipation rates are shown to be adequate. For the cross-dissipation rate, a constant normalized cross-dissipation rate is shown to be a poor model, and an improved model is proposed in which the normalized cross-dissipation rate changes sign across stoichiometric mixture fraction such that both lean and rich combustion are either simultaneously “back-supported” or simultaneously “front-supported”. Additional analysis is conducted to determine the sensitivity of the solutions of the one- and two-dimensional manifold equations to variations in these scalar dissipation profiles and indicates a relative insensitivity of the thermochemical state to the scalar dissipation rate profiles, especially compared to the sensitivity to the overall magnitude of the scalar dissipation rate or the inclusion of the often neglected cross-dissipation rate.