Two-dimensional Navier-Stokes Equations Research Articles

New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

AbstractThe paper deals with the stochastic two-dimensional Navier–Stokes equations for homogeneous and incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $$G\,\textrm{d}W$$ G d W , where W is a cylindrical Wiener process and G a bounded linear operator with range dense in the domain of $$A^\gamma $$ A γ , A being the Stokes operator. While it is known that existence of invariant measure holds for $$\gamma >1/4$$ γ > 1 / 4 , previous results show its uniqueness only for $$\gamma > 3/8$$ γ > 3 / 8 . We fill this gap and prove uniqueness and strong mixing property in the range $$\gamma \in (1/4, 3/8]$$ γ ∈ ( 1 / 4 , 3 / 8 ] by adapting the so-called Sobolevskiĭ-Kato-Fujita approach to the stochastic N–S equations. This method provides new a priori estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.

Performance evaluation of a scramjet engine utilizing varied cavity aft wall divergence with parallel injection in a reacting flow field

Abstract The effect of implications of the dual cavity with aft wall divergence in a parallel injection reacting flow has been numerically investigated. This research intends to emphasize the behaviour of the supersonic flow under varying divergence angles of the cavity aft wall. A two-dimensional Reynolds-Averaged Navier-Stokes (RANS) equation and an SST k-ω turbulence model with a single-step chemical reaction for hydrogen-air are utilized for the simulation. Followed by the strut injector, the cavities are positioned symmetrically inside the combustor. The bottom cavity aft wall divergence varies, whereas the top wall is mounted with a rectangular cavity. The performance of cavity locations is compared with the baseline DLR model. From the evaluation of numerical outcomes along with different cavity configurations, it has been noted that the 15-degree cavity divergence angle enhances the recirculation zone which leads to improved mixing performance. Also, the cavity improves the combustion stability by increasing the flow residence time. From this numerical analysis, it is associated that an almost 20 % reduction in combustor length is achieved, however, an 18 % rise in the pressure loss is noted because of emanating shock waves from the cavity edges.

On Efficient Method For Fractional-Order Two-Dimensional Navier-Stokes Equations

For the solution of fractional-order two-dimensional Navier-Stokes equations (FOTDNSEs), the current work proposes a novel variant of the Laplace Adomian decomposition approach with the Atangana-Baleanu fractional operator in the Caputo sense (ABCFO). In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the exact solution. Only two components are required for the new approximation analytical technique. When compared to other strategies, the current method is very simple, requires less calculation, and is extremely accurate.

Numerical simulations and theoretical analysis of the forward shock wave in a non-premixed air-breathing rotating detonation combustor

The forward shock wave induced by the detonation wave propagates in the inlet and restrains the air from being injected into the rotating detonation engine, and the stable continuous propagation of the detonation wave is affected. Numerical simulation and theoretical derivation of air flow velocity and detonation wave height were carried out in a rotating detonation combustor. The influence of the forward shock wave on air flow velocity and air flow velocity on detonation wave height was analyzed. A non-premixed air-breathing rotating detonation combustor was numerically simulated with the two-dimensional Navier–Stokes equations and a kerosene/air reaction model based on the RYrhoCentralFoam solver. Analytical equations were derived and validated to calculate the air flow velocity across the forward shock wave and the height of the detonation wave. The results show that the air flow velocity across the forward shock wave is inversely proportional to the detonation wave velocity, incoming flow temperature, and pressure ratio at the forward shock wave. In addition, the angle of the forward shock wave is inversely proportional to the velocity of the detonation wave and incoming flow and is proportional to the incoming flow temperature and the pressure ratio. The forward shock wave leads to a blocked zone of injection in the combustor, as the inflow velocity decreases or even reverses across the shock wave. As a result, the forward shock wave induced by the detonation wave restrains the recovery of the reactant fill zone, which has an influence on the height of the detonation wave.

Reconstruction of dynamic wind turbine wake flow fields from virtual Lidar measurements via physics-informed neural networks

Accurate characterisation of wind turbine wakes is important for the optimal design and operation of wind farms. However, current techniques for full-scale wind measurements are still limited to point characterisation. To address the research challenge in obtaining field characterisation of real-world wind turbine wakes, this work investigates the reconstruction of the dynamic wake flow fields based on a virtual turbine-mounted Lidar and physics-informed neural networks. Specifically, the wake flow field is reconstructed by fusing the sparse measurements with the two-dimensional Navier-Stokes equations without imposing any models for the unsteady wake. Different from supervised machine learning approaches which need the measured values for the quantities of interest in the first place, the proposed method can achieve the prediction of the wind velocity at new locations where there is no measurement available. The reconstruction performance is evaluated via high-fidelity numerical experiments and it is shown that the dynamic wind turbine wake flow fields are predicted accurately, where the main wake features, including the downwind development and crosswind meandering of the wake, are both captured. This work thus paves the way for investigating full-scale in situ wake flow dynamics in real-world wind energy sites.

Approximation of Two-Dimensional Time-Fractional Navier-Stokes Equations involving Atangana-Baleanu Derivative

This article addresses the two analytical methods, i.e., the new iterative transform method (NITM) and the homotopy perturbation transform method (HPTM), along with an Aboodh transform (AT), to approximate the nonlinear system of two-dimensional (2D) time-fractional Navier-Stokes (TFNS) equations. We take the time-fractional derivative in the form of Atangana-Baleanu (AB). The article's suggested examples examine the accuracy and efficacy of the proposed methods, while the graphs demonstrate their potential and effectiveness. The article also provides demonstrations of uniqueness and convergence. The aforementioned techniques are straightforward and support a high rate of convergence, which helps in understanding the dynamics of fractional nonlinear systems.

A MEEVC discretization for two-dimensional incompressible Navier-Stokes equations with general boundary conditions

In this work, we present a mass, energy, enstrophy and vorticity conserving (MEEVC) mixed finite element discretization for two-dimensional incompressible Navier-Stokes equations as an alternative to the original MEEVC scheme proposed in A. Palha and M. Gerritsma (2017) [5]. The present method can incorporate general boundary conditions. Conservation properties are proven. Supportive numerical experiments with both exact and inexact quadrature are provided.

Design, analysis, and testing of a prototype-scale latent heat thermal energy storage (LTES) system

A parametric analysis was performed to design a prototype-scale latent heat thermal energy storage (LTES) system using commercial grade hexahydrate calcium chloride as phase change material (PCM) in a staggered tube array configuration, placed horizontally. The study involved numerical analysis of melting and solidification processes of the PCM within the tube array, considering transient two-dimensional Navier-Stokes equations and a Realizable k-ɛ turbulence model to predict fluid flow and heat transfer. The enthalpy-porosity technique was used to model PCM melting and solidification. The study included a comparison between numerically predicted and experimentally obtained air temperatures at the tube array outlet, as well as PCM temperatures. This comparison revealed an excellent level of agreement. Additionally, the study compared the pressure drop within the array and the average peripheral heat transfer coefficient calculated from the numerical results to the well-established Zukauskas correlation. The resulting design successfully met the predefined performance criteria: achieving a cooling effect of 4 °C for a four-hour duration while maintaining a pressure drop of <100 Pa. The proposed prototype-scale tube array design can be efficiently cooled down and PCM frozen overnight. The energy storage density of the system falls within the range of 22 to 27 kWh/m3.

On the reconstruction of unknown driving forces from low-mode observations in the 2D Navier–Stokes equations

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.

Energy and exergy analysis of a laboratory-scale latent heat thermal energy storage (LTES) using salt-hydrate in a staggered tube arrangement

The energy and exergy analyses were performed for a laboratory-scale latent heat thermal energy storage (LTES) using hexahydrate calcium chloride (CC6) as phase change material (PCM) in a staggered tube array configuration, placed horizontally. The PCM melting and solidification process within the tube array was investigated by performing a numerical analysis using transient two-dimensional Navier-Stokes equations and Realizable k-ɛ turbulence model to predict flow and heat transfer. The enthalpy-porosity technique was applied to model PCM melting and solidification. The accuracy of the numerical model was validated against experimentally obtained data where the numerically predicted and measured air temperature at the tube array outlet and PCM temperatures were compared. Additionally, the pressure drop in the array and the average peripheral heat transfer coefficient calculated from the numerical results were compared to the well-known Zukauskas correlation (Zu). The results show that the melting and solidification process of PCM in the tubes of the array is asymmetric in nature, with the PCM melting taking slightly longer compared to the solidification process. It was also observed that the energy quantity was higher compared to the exergy quantity, as exergy considers entropy generation and accounts for irreversibility within the system.

A comparative study of latent heat thermal energy storage (LTES) system using cylindrical and elliptical tubes in a staggered tube arrangement

Numerical analysis was performed to compare the thermal and hydraulic performance of the elliptical and circular tube geometries in the prototype-scale latent heat thermal energy storage (LTES) system. A staggered tube array configuration was used for the analysis where the tubes were placed horizontally. Commercial grade hexahydrate calcium chloride (CC6) was selected as a phase change material (PCM) due to its phase change occurring at room temperature. The study included a numerical analysis of the melting and solidification processes of the PCM within the tube array for both tube shapes, employing transient two-dimensional Navier-Stokes equations and a Realizable k-ɛ turbulence model to predict fluid flow and heat transfer. The enthalpy-porosity technique was used to model PCM melting and solidification. Numerical predictions indicate that the thermal performance for both tube shapes is almost indistinguishable during the melting and freezing processes. However, due to the aerodynamic shape of the elliptical tube, the pressure drop across the horizontal elliptical tube array is approximately 80% lower compared to that of the circular geometry. This reduced pressure drop implies that less pumping power will be required to achieve the desired flow rate across the tube bank, leading to economic advantages for the horizontal elliptical tube array. The proposed design can be used as a dry cooling technique employing an Air Cooled Condenser (ACC) in a thermal power plant.

Convective thermal losses of long-term underground hot water storage

A case of underground long-term hot water storage is investigated numerically. The study is based on the unsteady two-dimensional Navier-Stokes equations in Boussinesq approximation applied to a closed cavern with time-dependent temperature boundary conditions on the walls. The problem formulated in a vorticity-stream function statement is solved by finite difference method (FDM) for high values of the Rayleigh number and for the Prandtl number of water. Streamlines, velocity and temperature fields are presented graphically for given moments of time. The evolution of the thermocline thickness in the mid-section of the cavern is slow and illustrates that the hot water zone occupies more than the half of the cavern even after 6 months period. The Nusselt number on the walls shows that the convective thermal losses are small and after certain period of time tend to decrease due to the diminished temperature difference at the walls. The influence of the fluid convection on the thermal losses is evaluated quantitatively, showing high seasonal thermal efficiency of the insulated hot water storage.

Stationary and Non-Stationary Periodic Flows Mathematical Modelling using Various Vortex Viscosity Models

Introduction. Mathematical modelling of currents is an urgent research topic in the field of hydrodynamics and oceanography. Despite ongoing research in the field of developing accurate and efficient numerical methods for solving Navier-Stokes equations that take into account vortex viscosity, the problems of accurate prediction and control of turbulence remain unresolved. The influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions also remains relevant. The aim of the study is to study the influence of linearized and quadratic bottom friction and two turbulence models on the numerical solution of stationary and non-stationary periodic flows. Special emphasis is placed on comparing numerical results with analytical solutions within the framework of using various models of bottom friction.Materials and Methods. The computational models used in this study are based on a simplified two-dimensional wave model and full three-dimensional Navier-Stokes equations. The classical model of shallow water motion and the 2D model without taking into account dynamic changes in the geometry of the reservoir surface are derived from a system of equations for a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects – nonlinearities caused by higher-order terms in the equations of motion, i. e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.Results. The results of modeling stationary and non-stationary periodic flows in a schematized rectangular basin using linearized bottom friction are presented. The influence of linearization on the numerical solution is investigated in comparison with analytical profiles using models calculating bottom friction in a quadratic formulation. In combination with quadratic bottom friction, two turbulence models are studied: the constant vortex viscosity and the Prandtl mixing length model. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modeling and analytical solutions averaged in depth.Discussion and Conclusion. New approaches to modelling and studying flows with variable vortex viscosity are proposed, including analysis of the influence of linearization and the use of various turbulence models. For the linearized and quadratic formulations of bottom friction, it is proved that the numerical results for the case of stationary flow show great similarity with analytical solutions, since the surface height is much less than the water depth and advection can be neglected. The numerical results for the unsteady flow also show a good agreement with the theory. Unlike analytical solutions, numerical modelling has minor deviations in the long run. The study of flows, within the framework of using various turbulence models, will make it possible to take into account the influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions. The results obtained make it possible to better understand and describe the physical processes occurring in shallow waters. This opens up new possibilities for applying mathematical modelling to predict and analyze the impact of human activities on the marine environment and to solve other problems in the field of oceanology and geophysics.

Direct numerical simulations of long-range infrasound propagation: Implications for source spectra estimation.

The evolution of observed dominant frequencies from a high-intensity infrasonic pulse with receiver range and stratospheric temperature is investigated using direct numerical simulations of the two-dimensional unsteady compressible Navier-Stokes equations. There is a high level of uncertainty in estimating source dominant frequencies based on received signals at sparse points on the ground. Nonlinear propagation effects in the ground-level thermospheric arrivals are found to significantly alter dominant frequency measurements compared to stratospheric arrivals with smaller amplitude sources. With a larger amplitude source, variations in observations are minimized as a result of nonlinear effects being ubiquitous across all atmospheric components of received signals but have a greater offset to the source dominant frequency. An approach to determine the source dominant frequency and minimize atmospheric variability is presented by calculating a source-to-receiver spectral transfer function averaged across the atmospheric states. This method reduces atmospheric variability in source frequency estimates within the pseudo-linear propagation regime and the average error to the known source frequency with a large amplitude source. The reduction of errors in source frequency estimates demonstrates the feasibility of using remote infrasound measurements as an indicator of source frequency and, in turn, the explosive yield of clandestine nuclear weapon test explosions.

Effect of confinement on the propagation patterns of lean hydrogen–air flames

The effect of confinement on the propagation of lean hydrogen–air flames in a Hele-Shaw cell, formed by two parallel plates separated by a small distance, is studied numerically. The influence of momentum loss on the flame structure and propagation rates is analyzed by examining two limits: (i) the limit of very small distance between the plates, for which a quasi-three-dimensional (quasi-3D) Darcy’s law-based approximation is used, and (ii) the limit of unconfined geometry, for which the full set of two-dimensional (2D) Navier-Stokes equations is considered. The influence of heat losses is also included. The chemistry is described by a one-step reduced kinetics with a simple model for transport properties. The results demonstrate that momentum loss primarily increases the flame surface area by elongating the cellular structures in a finger-like form, with a relatively small enhancement of the reactivity at the flame front, consistent with a mechanism of hydrodynamic nature. This elongation leads to approximately a 50% increase in flame speed compared to the absence of confinement. Conversely, heat losses are observed to flatten the large-scale continuous flame front and, if significant enough, can lead to the formation of isolated propagating flame cells of the two-headed or, ultimately, one-headed type. The present results are in good agreement with recent experiments.

Effect of discrete heating on the key parameters and entropy generation in a corrugated enclosure filled with hybrid nanofluid

This research work aims to study the effect of discrete heating on natural convective heat transfer, fluid flow, and entropy generation within a corrugated enclosure filled with water-based molybdenum disulfide (MoS2)–silicon dioxide (SiO2) hybrid nanofluid. In the study, various heights of the discrete heaters in terms of aspect ratios ( AR = 0.2 , 0.4 , 0.6 , 0.8 ) are considered for various Rayleigh numbers ( Ra = 10 3 − 10 6 ) with varying nanoparticle volume fractions ( ϕ = 0 % − 4 % ). Stream function-vorticity formulations of two-dimensional (2D) Navier-Stokes equations and energy equation are used as the governing equations, and a higher-order compact (HOC) finite difference scheme is employed to discretize those equations. The results obtained from the current developed code are first validated with the existing numerical and experimental data. Finally, the present computed results are studied, both qualitatively and quantitatively, in terms of streamlines, isotherms, entropy generations, average Nusselt numbers, and Bejan numbers. At Ra = 10 6 and the aspect ratio of heater length AR = 0.2 , a different and interesting flow and heat transfer phenomenon is observed. The results show that an increase in the aspect ratio of heater length from 0.2 to 0.8 decreases the average Nusselt number and total entropy generation by 57.49% and 13.76%, respectively, for Ra = 10 6 and ϕ = 4 % . In addition, the effect of nanoparticle concentration on total entropy generation increases up to 73.40% when the aspect ratio of heater length increases to 0.8.

Numerical simulations of two-dimensional incompressible Navier-Stokes equations by the backward substitution projection method

The backward substitution method is a newly developed meshless method that has been used for the simulation of many problems in science and engineering with high accuracy and efficiency. In this paper, we explore the feasibility of employing the backward substitution method for simulating two-dimensional incompressible flows. Two non-increment pressure correction projection methods are considered to decompose the original velocity and pressure coupling system into two boundary value problems of intermediate velocity and pressure. Then, two boundary value problems are solved by the backward substitution method in each time iteration step. Five numerical examples are provided to demonstrate the accuracy, computational efficiency and convergence of the method. Comparisons with some existing meshless methods verify the method's advantages and potential applications to engineering.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

Numerical simulation of acoustic streaming in standing waves

This study presents a numerical investigation of acoustic streaming motion (of the Rayleigh type) in a compressible gas inside two-dimensional rectangular enclosures. To numerically study the effects of the sound field intensity on the formation process of streaming structures, we propose to discretize the fully compressible form of the two-dimensional Navier-Stokes equations using a high-order (formally greater or equal to fourth-order) accurate numerical scheme in both space and time. The proposed numerical solver utilizes high-order compact schemes along each spatial dimension combined with a filtering procedure when it is necessary. Acoustic standing waves are excited inside the enclosures and the resulting acoustic streaming patterns are investigated for low and high-intensity waves, in both linear and nonlinear regimes following closely the work of Daru et al. (2013). An extended investigation indicates that without the incorporation of an appropriate filter, the application of the high-order compact schemes in case of fast streaming results in spurious oscillations which inhibit their applicability. Following the recent relevant literature, the numerical simulations performed demonstrate the ability of the proposed numerical approach to reproduce efficiently and robustly the transitions from regular acoustic streaming to irregular streaming and the relevant phenomena confirming results that have been previously presented.

On variable viscosity and enhanced dissipation

In this article we consider the two-dimensional Navier–Stokes equations with variable viscosity depending on the vertical position. As our main result we establish linear enhanced dissipation near the non-affine stationary states replacing Couette flow. For instance, these shear flows may grow exponentially. Moreover it turns out that, in contrast to the constant viscosity case, decreasing viscosity leads to stronger enhanced dissipation and increasing viscosity leads to weaker dissipation.