Two-dimensional Direct Numerical Simulations Research Articles

High-speed microactuation in a supersonic dual-stream jet flow

Supersonic shear layers experience instabilities that generate significant adverse effects; in complex configurations, these instabilities have global impacts as they foster compounding complications with other independent flow features. We consider the flow near the exit of a dual-stream rectangular nozzle. The supersonic core and sonic bypass streams mix downstream of a splitter plate trailing edge (SPTE) just above an adjacent deck. We perform two-dimensional direct numerical simulations with laminar inflow conditions to parametrically explore the influence of active flow control, considering various actuation angles and locations. The goal is to alleviate the prominent tone associated with vortices shed at the SPTE; these vortices initiate an unsteady shock system that affects the entire flow field through a shock-induced separation and the downstream evolution of plume shear layers. Resolvent analysis is performed on the baseline flow. The identified optimal response guides the placement of steady-blowing actuators. Since the resolvent analysis fundamentally investigates the input–output dynamics of a system, it is also utilized to uncover actuation-induced changes in the forcing–response dynamics. Spectral analysis shows that the baseline flow fluctuating energy is concentrated in the shedding instability. Actuating at optimal angles based on location disperses this energy into various flow features; this affects the shedding itself, and the structure and unsteadiness of the shock system, and thus the response of the deck and nozzle wall boundary layers and the plume. The resolvent analysis indicates, and Navier–Stokes solutions confirm, that favourable control is obtained by either indirectly or directly mitigating the baseline instability.

Spontaneous ignition and flame propagation in hydrogen/methane wrinkled laminar flames at reheat conditions: Effect of pressure and hydrogen fraction

Direct numerical simulations (DNS) with detailed chemical kinetics and chemical explosive mode analysis (CEMA) are performed to investigate the effect of operating pressure and hydrogen–methane blending on laminar wrinkled flames at reheat combustion conditions. Building upon previous three-dimensional DNS datasets of reheat hydrogen flames, the present work considers two-dimensional configurations that render computationally-feasible simulations of high-pressure combustion and significantly more complex hydrocarbon chemistry. A geometrically-simplified representation of the reheat combustor is adopted and the thermodynamic states considered in the simulations are carefully chosen to mimic gas turbine operation conditions, which are constrained to achieve a target flame position and flame temperature. The two-dimensional DNS results are first assessed against the available three-dimensional data and then analyzed to extract the main trends exhibited by the reheat combustion process with respect to the fraction of the fuel consumed by spontaneous ignition and flame propagation, for increasing pressures and hydrogen fractions. Due to significant differences in the characteristic features of the velocity and thermal boundary layers between the three-dimensional and the two-dimensional configurations, a different global flame shape is observed (V-shaped flame vs W-shape flame) together with a ∼10% bias in the fraction of fuel consumed by spontaneous ignition. Crucially, results from the scaling studies reveal very clear trends with a significant decrease in the fuel consumption by spontaneous ignition for increasing pressures and hydrogen fractions, at the conditions investigated. Finally, this study highlights the important role that first-principle direct numerical simulations, even when conducted in computationally affordable two-dimensional simplified geometrical configurations, can play in obtaining detailed insights and quantitative trends useful for the characterization of complex reactive flows.Novelty and significanceThe reheat burners of the two-stage sequential gas turbine combustors are designed to stabilize flames primarily due to spontaneous ignition. However, prior numerical simulations revealed the presence of an additional assisted-ignition mode aided by recirculation zones. In this study, we used practically feasible two-dimensional direct numerical simulations of premixed hydrogen–air mixtures under lean conditions to quantify the roles of the two flame stabilization modes at different operating pressures. Achieving fundamental understanding of the effect of pressure and hydrogen fraction on flame stabilization is crucial to the development of two-stage sequential combustion and the present two-dimensional calculations provide important new insights. We show that at high pressures, both modes consume fuel to a similar extent. We also investigated the effect of methane blending on flame stabilization. This study also discusses how two-dimensional direct numerical simulations can be leveraged in the design optimization of reheat burners at low technology readiness levels.

Dynamics of slowly propagating flames: Role of the Rayleigh-Taylor instability

Environmental concerns have motivated the development of alternate fuels and refrigerant working fluids that have low global warming potential. Ammonia (NH3) is a candidate zero-carbon fuel and hydrofluorocarbons (HFCs) such as R-32 and R-1234yf are being adopted as refrigerants. Upon mixing with air, these compounds can sustain flames that are slowly propagating i.e., with laminar flame speeds less than 10 cm/s. These flames are referred to as slow as they are affected by buoyancy-induced flow and radiation heat loss, in contrast to typical hydrocarbon-fueled flames. In this study, we investigate the flame dynamics of slowly propagating refrigerant/air flames, with a focus on the effect of the Rayleigh-Taylor (RT) instability. Dispersion relations, that characterize the range of unstable wavelengths and their growth rates during early stages of instability growth, are derived from two-dimensional direct numerical simulations (DNS, including detailed chemistry and transport) of RT-unstable R-32/air flames. These results are compared to predictions from reduced-order theoretical models. DNS are also performed to study the long-term evolution of such flames and to quantify flame speed enhancements. Results showed that RT-unstable flames can propagate up to 6–7 times the laminar flame speed. Although slow from a flamelet point of view, these flames propagate much faster at larger scales. These results have implications on developing metrics to assess explosion risk, and minimize it while storing, transporting, and utilizing these compounds. (225 words)

Physics-Informed Neural Networks for Prediction of a Flow-Induced Vibration Cylinder

Abstract Flow-induced vibration (FIV) is a common phenomenon in ocean engineering for subsea structures with circular-shaped cross sections. A large amount of computational resources or experimental efforts are required to predict or measure the complicated motions and flow field surrounding a circular cylinder undergoing vortex-induced vibration (VIV). Physics-informed neural networks (PINNs) are powerful deep learning techniques for solving governing partial differential equations (PDEs) of dynamic systems as an alternative to complex numerical methods. In the present study, a framework is built employing PINNs for solving the Navier–Stokes equations to predict flows past an FIV cylinder using sparsely distributed spatiotemporal data inside the domain. The training process involves minimizing the supervised loss of flow data at these sparse points and the residuals of the governing PDEs. For the training of the PINN model, a moving frame around an FIV cylinder is used to collect the training flow data from two-dimensional direct numerical simulation results at a low Reynolds number. The structural displacements of the cylinder are also implemented in the residuals of the equations of the developed PINN. The performance of the PINN is evaluated by comparing the predicted contours of the surrounding flow velocities with the training data. The hydrodynamic forces prediction is achieved using the PINN-obtained flow field predictions combined with the force partitioning method.

Thermoviscous dissipation of nonlinear acoustic waves in channels with wavy walls.

We derive a nonlinear acoustic wave propagation model for analyzing the thermoviscous dissipation of nonlinear acoustic waves in narrow pores with wavy walls using the boundary layer theory. As a nonlinear acoustic wave propagates in a pore, the wave-steepening effect competes with the bulk dissipation, as well as the thermoviscous heat transfer and shear from the pore walls. Due to thermoviscous dissipation, the wave thickness increases beyond the weak shock thickness scale. Using the weak shock thickness scale, we obtain dimensionless linear and nonlinear model wave equations governing the shock-wall interactions. We also perform two-dimensional shock-resolved direct numerical simulation of the wave propagation inside the pores and compare the results with model equations. The direct numerical simulation and model calculations show that, for flat walls and shock strength parameter ϵ, the dimensional wall heat-flux and shear scale as ϵ. For wavy walls, the scaling becomes ϵ3/2-n(k) where k is the wall-waviness wavenumber and the exponent n increases from 0.5 for k = 0 to n(k)≈0.65 for k = 10, n(k)≈0.75 for k = 20, and n(k)≈0.85 for k = 40. Furthermore, we show that both the dimensionless scaled wall shear and wall heat-flux decrease with increasing k.

Towards Direct Numerical Simulations of Reactivity-Controlled Compression Ignition Engine Using n-Octanol/Ethanol Fuel Blends

AbstractThe results of a two-dimensional direct numerical simulation of a lean n-octanol-ethanol fuel blend under Reactivity Controlled Compression Ignition (RCCI) conditions are presented in this paper. Stratified temperature and high reactivity fuel fields of Gaussian, bi-modal, and log-normal distributions are studied for uncorrelated and correlated scenarios. The pimple loop is made to run twice to achieve compression heating. A chemical mechanism with 171 species and 734 reactions was developed to capture autoignition characteristics and flame propagation reasonably well. Diagnosing techniques published in the literature are used to determine whether the flame fronts are spontaneously propagating or not. As reported previously for other fuel blends under RCCI conditions, both deflagration and spontaneous ignition flame fronts are observed to co-exist. Gaussian, bi-modal, and log-normal fields respectively move towards a spontaneously igniting mode. Correlating temperature and high reactivity fuel fields not only makes combustion more spontaneously igniting but also more premixed. The analysis reveals the sensitivity of the DNS results with respect to the initial conditions which accordingly should be chosen with care.

The boundary layer instability beneath internal solitary waves and its sensitivity to vortex wakes

We investigated the stability of the bottom boundary layer (BBL) beneath periodic internal solitary waves (ISWs) of depression over a flat bottom through two-dimensional direct numerical simulations. We explored the convective versus absolute/global nature of the BBL instability in response to changes in Reynolds number, and the sensitivity of the instability to seeding noise in the front of the ISW – spanning laboratory to geophysical scales. The BBL was laminar at $Re_{ISW}=90$ and convectively unstable at $Re_{ISW}=300$ . At laboratory-scale $Re_{ISW}=300$ , the convective wave packet was periodically amplified by each successive ISW, until vortex shedding occurred. The associated noise-amplification behaviour potentially explains the discrepancies of the critical $Re_{ISW}$ between the lock–release laboratory experiments and our Dubreil–Jacotin–Long-initialized numerical simulations as the result of the difference in background noise. Instability energy decreased under the front shoulder of the ISW, analogous to flow relaminarization under a favourable pressure gradient. At geophysical-scale $Re_{ISW}=900$ , the BBL was initially convectively unstable, and then the instability tracked with the ISW, appearing phenomenologically similar to a global instability. The simulated initial convective instability at both $Re_{ISW}=300$ and $Re_{ISW}=900$ is in agreement with local linear stability analysis which predicts that the instability group speed is always lower than the ISW celerity. Increased free stream perturbations in front of the ISW and larger $Re_{ISW}$ shift the location of vortex shedding (and enhanced bed shear stress) beneath the wave, closer to the ISW trough, thereby potentially changing the location of maximum sediment resuspension, in agreement with field observations at higher $Re_{ISW}$ .

Convection driven by a nonuniform radiative internal heat source in a cavity: Example of medical isotope production in liquid targets

In the present study we use two-dimensional direct numerical simulations (DNS) to understand the coupled heat transfer to fluid flow in a liquid target for the production of nuclear medicines. Fluid motion is driven by buoyancy created by heat generated by a proton beam. The internal heat source has a gaussian distribution in the vertical direction and a rapidly growing intensity in the horizontal direction until it reaches a range at the Bragg peak where the heating drops to zero. The structure of the heating imposes two convective cells, separated at the location of the range. We solve the governing fluid flow and energy equations in a square cavity subject to highly nonuniform internal heating generated by the energy deposition of a proton beam. While most studies of convection driven by an internal heat source in a fluid layer have been focused on a uniform heating of the fluid, our study shows that the nonuniformity in the heat source has important implications for the temperature and flow fields, the boundary heat fluxes, and the growth of convective instabilities in the flow. Interestingly, the scalings of the maximum and averaged temperatures with the Rayleigh number compare similarly to previously found power laws for uniformly heated fluid layers. At higher power levels, the layer of fluid near the top cold boundary becomes convectively unstable via Rayleigh–Taylor instabilities. By comparing the rate of growth of these instabilities to their rate of advection to the boundaries of the cavity, a model is developed that predicts the instability of the convective cells for different values of the range of the beam and the Rayleigh number. Crucially, we demonstrate that the disturbances in the production of isotopes due to convective instabilities and the design of the cooling system is dependent on the location of the Bragg peak and must be considered in design of future generation of this class of target.

The Impact of Roughness and Partitions in Thermal Convection to Enhance the Heat Transport

Abstract In this investigation, we explore the profound impact of adiabatic partitions on heat transport within a square Rayleigh-Bénard (RB) convection enclosure characterized by surface roughness on the lower hot and upper cold plates. The surface irregularities take the form of a rectangular base and a triangular apex. Our study employs two-dimensional direct numerical simulations spanning the Rayleigh number (Ra) range of 106–108 and a Prandtl number (Pr) of 1. Adiabatic partitions, strategically positioned between successive roughness, serve as the focal point of our exploration. Remarkably, our findings reveal a substantial enhancement in heat transport with the introduction of a partition board between consecutive roughness elements. As we escalate the number of partitions from one to four, the heat flux experiences a notable augmentation, reaching 2.3 times that of the classical square RB configuration. This enhancement is attributed to the breakdown of large-scale rolls into multiple rolls, a phenomenon intensified by the increased partition height. Further intriguing observations unfold as we investigate the interplay of surface roughness and partitions. Configurations featuring roughness with partitions exhibit an impressive 2.7 times heat flux enhancement compared to the classical square RB setup. The complex interplay of heat transport improvement is closely connected to optimizing the distance between the conduction plate and the partition. Through meticulous analysis, we identify that the optimal gap facilitates heightened local velocity, effectively thinning the thermal boundary layer and consequently augmenting the overall heat flux. In essence, our study sheds light on the synergistic effects of adiabatic partitions and surface roughness in the context of RB convection. The observed enhancements in heat transport underscore the potential for tailored design strategies involving partitions and surface modifications to optimize thermal performance in diverse applications.

Two-dimensional direct numerical simulation study of multicomponent mixing with phase transition in a transcritical shear layer

In this paper, we investigate two-dimensional direct numerical simulations of a transcritical shear layer. Three configurations are chosen, which are distinguished by the level of presence of two-phase phenomena. The thermodynamic model is based on a cubic equation of state. It was extended for multicomponent mixtures, and it is able to account for vapor–liquid equilibrium. The thermodynamic modeling with phase-transition is validated using experimental data from the literature. Special focus is put on the effect of the density gradient and the density changes caused by phase-transition on the development of the turbulent shear layer and the associated mixing. In addition to this, the vorticity distribution and the components of its transport equation are analyzed and compared for the different configurations.

Towards the understanding of convective dissolution in confined porous media: thin bead pack experiments, two-dimensional direct numerical simulations and physical models

We consider the process of convective dissolution in a homogeneous and isotropic porous medium. The flow is unstable due to the presence of a solute that induces a density difference responsible for driving the flow. The mixing dynamics is thus driven by a Rayleigh–Taylor instability at the pore scale. We investigate the flow at the scale of the pores using Hele-Shaw type experiment with bead packs, two-dimensional direct numerical simulations and physical models. Experiments and simulations have been specifically designed to mimic the same flow conditions, namely matching porosities, high Schmidt numbers and linear dependency of fluid density with solute concentration. In addition, the solid obstacles of the medium are impermeable to fluid and solute. We characterise the evolution of the flow via the mixing length, which quantifies the extension of the mixing region and grows linearly in time. The flow structure, analysed via the centreline mean wavelength, is observed to grow in agreement with theoretical predictions. Finally, we analyse the dissolution dynamics of the system, quantified through the mean scalar dissipation, and three mixing regimes are observed. Initially, the evolution is controlled by diffusion, which produces solute mixing across the initial horizontal interface. Then, when the interfacial diffusive layer is sufficiently thick, it becomes unstable, forming finger-like structures and driving the system into a convection-dominated phase. Finally, when the fingers have grown sufficiently to touch the horizontal boundaries of the domain, the mixing reduces dramatically due to the absence of fresh unmixed fluid. With the aid of simple physical models, we explain the physics of the results obtained numerically and experimentally. The solute evolution presents a self-similar behaviour, and it is controlled by different length scales in each stage of the mixing process, namely the length scale of diffusion, the pore size and the domain height.

Flow-induced vibration of two tandem square cylinders at low Reynolds number: transitions among vortex-induced vibration, biased oscillation and galloping

The flow-induced vibrations (FIVs) of two identical tandem square cylinders with mass ratio m* = 3.5 at Reynolds number Re = 150 are investigated through two-dimensional direct numerical simulation (DNS) and linear stability analysis over a parameter range of spacing ratio 1.5 ≤ L* ≤ 5 and reduced velocity 3 ≤ Ur ≤ 34. Three kinds of FIV responses, namely vortex-induced vibration (VIV), biased oscillation (BO) and galloping (GA), are identified. The FIVs are then further classified into the branches of initial VIV (IV), resonant VIV (RV and RV′), flutter-induced VIV (FV), desynchronized VIV (DV), VIV developing from GA (GV), transitional state between VIV and GA (TR), BO and GA based on the characteristics of the vibration responses. The transitions among different FIV branches are examined by combining the DNS with linear stability analysis, where the transition boundaries among the VIV, BO and GA branches over the concerned parameters are identified on the branch maps. The transition from IV to RV or RV′ is found to be related to the unstable wake mode, while the FV, transiting from RV or RV′, is induced by the unstable structural factor in the wake-structure mode. The structural instability is considered as the physical origin of GA, whereas the mode competition between unstable wake and structure leads to DV, GV and TR, and thus delays the appearance of GA. The transition from DV to BO with biased equilibrium position, accompanied by the even-order harmonic frequencies, is essentially induced by the symmetry breaking bifurcation.

Influence of slit asymmetry on blow-off and flashback in methane/hydrogen laminar premixed burners

One way to decarbonize the heat production sector is to gradually replace natural gas with hydrogen. The design of laminar premixed burners capable of operating with both methane and hydrogen is however challenging as these fuels have drastically different burning properties, which narrows the operating range. While methane flames face limitations in stabilization due to blow-off at high power, hydrogen flames tend to be susceptible to flashback at low power. This study investigates the effects of slits symmetry breaking on the blow-off of methane flames and flashback of hydrogen flames through two-dimensional direct numerical simulations of canonical asymmetrical slit configurations, revealing uneven interactions between the main openings and smaller auxiliary slits. The equations governing the reactive flow dynamics are coupled to a heat transfer solver in the solid phase to elucidate the thermal and hydrodynamic mechanisms determining the operability limits. Viscous dissipation in the small auxiliary slits together with a substantial preheating of the fresh gases by heat redistribution through the solid phase is found to govern flame stabilization in asymmetrical geometries, improving blow-off resistance of methane-air flames. Then, flashback of hydrogen-air flame is found to be driven by the competition between (i) the mass flow rate distribution between the main and auxiliary slits, (ii) preheating of the gas through the auxiliary slits and (iii) the ability of the main and auxiliary slits to quench the flame. The interplay of these phenomena gives rise to complex behaviors, wherein asymmetrical configurations could exhibit significantly enhanced resistance to flashback compared to symmetrical geometries. This conclusion, verified for different burner thicknesses and slit spacing, may be used to guide the design of fuel-flexible laminar burners.Novelty and significance statement1. Analyzing the effect of slits symmetry breaking on blow-off of methane flames and flashback of hydrogen flames for multi-perforated laminar burners.2. Defining a strategy for wall heat flux redistribution to assess the asymmetrical interaction between the main and auxiliary flames through the flame-holder.3. Demonstrating and explaining the positive impact of symmetry breaking on the blow-off resistance of methane flames.4. Highlighting the substantial and non-monotonous impact of symmetry breaking on hydrogen flame flashback.5. Identifying the interplay of thermal and hydrodynamic mechanisms in asymmetrical slits leading to improvement of flashback resistance for hydrogen-air flames compared to symmetrical configurations.

Multistability of elasto-inertial two-dimensional channel flow

Elasto-inertial turbulence (EIT) is a recently discovered two-dimensional chaotic flow state observed in dilute polymer solutions. Two possibilities are currently hypothesized to be linked to the dynamical origins of EIT: (i) viscoelastic Tollmien–Schlichting waves and (ii) a centre-mode instability. The nonlinear evolution of the centre mode leads to a travelling wave with an ‘arrowhead’ structure in the polymer conformation, a structure also observed instantaneously in simulations of EIT. In this work we conduct a suite of two-dimensional direct numerical simulations spanning a wide range of polymeric flow parameters to examine the possible dynamical connection between the arrowhead and EIT. Our calculations reveal (up to) four coexistent attractors: the laminar state and a steady arrowhead regime (SAR), along with EIT and a ‘chaotic arrowhead regime’ (CAR). The SAR is stable for all parameters considered here, while the final pair of (chaotic) flow states are visually very similar and can be distinguished only by the presence of a weak polymer arrowhead structure in the CAR regime. Analysis of energy transfers between the flow and the polymer indicates that both chaotic regimes are maintained by an identical near-wall mechanism and that the weak arrowhead does not play a role. Our results suggest that the arrowhead is a benign flow structure that is disconnected from the self-sustaining mechanics of EIT.

Low-Prandtl-number effects on global and local statistics in two-dimensional Rayleigh–Bénard convection

We present global and local statistical properties of turbulent Rayleigh–Bénard (RB) convection at low Prandtl numbers in this work. A series of high resolution two-dimensional (2D) direct numerical simulations are carried out in a square box for the Prandtl number ranges 0.005≤Pr≤0.07 and 0.01≤Pr≤0.15 at Rayleigh numbers Ra=107 and Ra=108, respectively. The global heat and momentum transport expressed as Nusselt number Nu and Reynolds number Re are found to scale as Nu∼Pr0.14 and Re∼Pr−0.82 for Ra=107, and Nu∼Pr0.11Re∼Pr−0.93 for Ra=108. The local velocity fluctuation at the cell center shows larger amplitudes at lowered Pr, indicating a stronger turbulence in the bulk. The magnitudes of kinetic and thermal energy dissipation rates in the bulk also increase with the decreasing of Pr, due to the intensified velocity gradient and larger thermal diffusivity, respectively. In the cell central region, probability density functions (PDFs) of velocity show a bimodal distribution, and it approaches the Gaussian distribution at higher Pr, while the PDFs of temperature display a stretched exponential shape with intermittent behavior. The kinetic energy spectra further reveal that the velocity cascade follows the Bolgiano–Obukhov scaling in the bulk of the convective flow.

Thermal diffusion, exhaust gas recirculation and blending effects on lean premixed hydrogen flames

Thermodiffusively-unstable lean premixed hydrogen flames are investigated using two-dimensional direct numerical simulation employing finite-rate chemical kinetics. Three databases are generated focussing on the inclusion of the Soret effect, the recirculation of exhaust gas, and blending with methane. A simple rescaling of a classic thermal diffusion model is presented and shown to mimic multicomponent diffusion with very low computational cost and little-to-no loss in accuracy. It is also shown that a previously developed model for mean local flame speeds in lean premixed hydrogen flames [1] can still be used provided Soret effects are taken into account in one-dimensional calculations. The addition of exhaust gas to the unburned mixture is found to enhance thermodiffusive instability; the primary mechanism for this was shown to be the highly-efficient third-body nature of water, with the reduction of adiabatic flame temperature a second-order effect. Again, the existing mean local flame speed model proved sufficient. Finally, blending with methane was found to reduce the thermodiffusive response of the flame, more so than the existing model suggests, despite adjustment of the fuel Lewis number; an adapted model is presented to account for this.

Optimizing flow control with deep reinforcement learning: Plasma actuator placement around a square cylinder

This study introduces a deep reinforcement learning-based flow control approach to enhance the efficiency of multiple plasma actuators on a square cylinder. The research seeks to adjust the control inputs of these actuators to diminish both drag and lift forces on the cylinder, ensuring flow stability in the process. The proposed model uses a two-dimensional direct numerical simulation of flow past a square cylinder to represent the environment. The control approach involves adjusting the AC voltage across three specific configurations of the plasma actuators. Initially tested at a Reynolds number (ReD) of 100, this strategy was later applied at ReD of 180. We observed a 97% reduction in the mean drag coefficient at ReD = 100 and a 99% reduction at ReD = 180. Furthermore, the findings suggest that increasing the Reynolds number makes it harder to mitigate vortex shedding using plasma actuators on just the cylinder's rear surface. However, an optimized configuration of these actuators can fully suppress vortex shedding under the proposed control scheme.

Scaling relations in quasi-static magnetoconvection with a strong vertical magnetic field

The scaling law for the horizontal length scale $\ell$ relative to the domain height $L$ , originating from the linear theory of quasi-static magnetoconvection, $\ell /L \sim Q^{-1/6}$ , has been verified through two-dimensional (2-D) direct numerical simulation (DNS), particularly at high values of the Chandrasekhar number ( $Q$ ). This relationship remains valid within a specific flow regime characterized by columnar structures aligned with the magnetic field. Expanding upon the $Q$ -dependence of the horizontal length scale, we have derived scaling laws for the Nusselt number ( $Nu$ ) and the Reynolds number ( $Re$ ) as functions of the driving forces (Rayleigh number ( $Ra$ ) and $Q$ ) in quasi-static magnetoconvection influenced by a strong magnetic field. These scaling relations, $Nu \sim Ra/Q$ and $Re \sim Ra Q^{-5/6}$ , have been successfully validated using 2-D DNS data spanning a wide range of five decades in $Q$ , ranging from $10^5$ to $10^9$ . The successful validation of the relations at large $Q$ values, combined with our theoretical analysis of dissipation rates and the incorporation of the horizontal length scale's influence on scaling behaviour, presents a valid approach for deriving scaling laws under various conditions.

Wave statistics and energy dissipation of shallow-water breaking waves in a tank with a level bottom

The present study focuses on two-dimensional direct numerical simulations of shallow-water breaking waves, specifically those generated by a wave plate at constant water depths. The primary objective is to quantitatively analyse the dynamics, kinematics and energy dissipation associated with wave breaking. The numerical results exhibit good agreement with experimental data in terms of free-surface profiles during wave breaking. A parametric study was conducted to examine the influence of various wave properties and initial conditions on breaking characteristics. According to research on the Bond number ( $Bo$ , the ratio of gravitational to surface tension forces), an increased surface tension leads to the formation of more prominent parasitic capillaries at the forwards face of the wave profile and a thicker plunging jet, which causes a delayed breaking time and is tightly correlated with the main cavity size. A close relationship between wave statistics and the initial conditions of the wave plate is discovered, allowing for the classification of breaker types based on the ratio of wave height to water depth, $H/d$ . Moreover, an analysis based on inertial scaling arguments reveals that the energy dissipation rate due to breaking can be linked to the local geometry of the breaking crest $H_b/d$ , and exhibits a threshold behaviour, where the energy dissipation approaches zero at a critical value of $H_b/d$ . An empirical scaling of the breaking parameter is proposed as $b = a(H_b/d - \chi _0)^n$ , where $\chi _0 = 0.65$ represents the breaking threshold and $n = 1.5$ is a power law determined through the best fit to the numerical results.

Non-Oberbeck–Boussinesq effects on a water-filled differentially heated vertical cavity

In this study, we examined non-Oberbeck–Boussinesq (NOB) effects on a water-filled differentially heated vertical cavity through two-dimensional direct numerical simulations. The simulations encompassed a Rayleigh number (Ra) span of 107–1010, temperature difference (Δθ̃) up to 60 K, and a Prandtl number (Pr) fixed at 4.4. The center temperature (θcen) was found to be independent of Ra and to increase linearly with Δθ̃, as presented by θcen≈1.18×10−3 K−1Δθ̃. The thermal boundary layer (BL) thicknesses near the hot and cold walls (λ¯hθ and λ¯cθ, respectively) are found to scale as λ¯h,cθ∼Raγ λ¯h,c, where the scaling exponent γ λ¯h,c ranges from −0.264 to −0.262. For more detail, the scaling exponent γ λ¯h displays an increasing trend, while γ λ¯c demonstrates a decreasing trend. However, the sum of the hot and cold thermal BL thicknesses was found to be constant at a fixed Ra in the presence of NOB effects. Our detailed investigation of the Nusselt number (Nu) and Reynolds number (Re) revealed that Nu∼Ra0.258 and Re∼Ra0.364, showing insensitivity to NOB effects. These exponents were smaller than those for Rayleigh–Bénard convection. The NOB modifications on Nu and Re were less than 1.2% and 2.5%, respectively, even at Δθ̃=60 K. Our results also revealed that key parameters such as θcen and normalized ratios [(λ¯NOBθ/λ¯OBθ)h,c, NuNOB/NuOB, and ReNOB/ReOB] exhibit universal correlations with Δθ̃. Remarkably, these relationships are consistent across varying Ra values. This observation underscored the influence of NOB effects on these parameters could be confidently forecasted using just the temperature difference (Δθ̃) for Ra∈[107,1010].