Full Navier Research Articles

A Perspective on the Process and Turbomachinery Design of Compressed Air Energy Storage Systems

Abstract The present paper will describe the Baker Hughes experience in the development of the turbomachinery equipment for Hydrostor's advanced compressed air energy storage (A-CAES) system. At the core of a compressed air energy storage (CAES) plant, there is an air compressing system, followed by an air expander used to recover the stored energy. To achieve a reliable and effective solution, the expander is obtained from the architecture of Baker Hughes steam turbines, which was adapted to match the specific process needs. The criticalities that were addressed and solved to derive the expanders from the original steam turbine are presented. Specifically, the paper describes the activities performed to optimize the inlet and exhaust sections of each segment, the development of the blades for high atmospheric volume flow, and the implications that thermal transients have on the machine reliability. The inlet and exhaust sections were arranged according to the layout constraints, which were set to mitigate the effects of the thermal stresses and to reduce the weight to facilitate the machine transportation, while maintaining high aero-performance. As for the expander, a single-body configuration was selected to optimize capital expenditures and reduce leakage to atmosphere. A new set of blades derived from Baker Hughes's Steam Turbine stages were developed. This new stage is characterized by rotating blades with high radius ratio (HRR); therefore, an optimization strategy was adopted to include the mechanical constraints from the beginning of the design cycle and obtain a final geometry that can be used with different flow path and operating conditions. Finally, the three-dimensional full Navier Stokes Computational fluid dynamics analysis was used to assess the performance of the new stages in nominal and off-design conditions. A detailed analysis of the thermal transient of the expander parts with finite element analysis methods was performed to assess the life expectations of the equipment. The finite element analysis results are discussed in the paper to show the capability of the machine to sustain an extremely fast start up sequence. Compressor train is characterized by a multiple body configuration. A detailed optimization was performed to improve the efficiency and the availability of the selected solution. The low-pressure axial compressor discharge section has been optimized to reduce losses and a detailed study of rotor has been performed to evaluate the capability to withstand a high number of start and stop cycles.

Hydrodynamics of multicomponent vesicles: A phase-field approach

In this paper, we introduce a thermodynamically-consistent phase-field model to investigate the hydrodynamics of inextensible multicomponent vesicles in various fluid flows with inertial forces. Our model couples the fluid field, surface concentration field representing chemical species on the membrane, and vesicle dynamics, while enforcing global area and volume constraints through a Lagrange multiplier method. Specifically, we employ full Navier–Stokes equations for the fluid field, the Cahn–Hilliard equations for the species concentration on the membrane, a nonlinear advection–diffusion equation to describe the evolution of the vesicle membrane, and an additional equation to enforce local inextensibility. We utilize a residual-based variational multiscale method for the Navier–Stokes equations and a standard Galerkin finite element framework for the remaining equations. The PDEs are solved using an implicit, monolithic scheme based on the generalized-α time integration method. We extend previous models for homogeneous vesicles (Aland et al., 2014; Valizadeh and Rabczuk, 2022) to multicomponent vesicles, introducing Cahn–Hilliard equations while maintaining thermodynamic consistency. Additionally, we employ isogeometric analysis (IGA) for higher accuracy. We present a variety of two-dimensional numerical examples, including multicomponent vesicles in shear flow and Poiseuille flow with and without obstructions, using the resistive immersed surface method to handle obstructions. Furthermore, we provide three-dimensional simulations of multicomponent vesicles in Poiseuille flow.

Modelling the Flow in the Utah FORGE Wells Disrete Fracture Network

The focus of this paper is the efficient numerical solution of the fluid flow in the Utah Frontier Observatory for Research in Geothermal Energy (FORGE) reservoir. In this study, the public data available for Discrete Fracture Networks (DFN) around well 58-32 is used to represent the DFN. In this research, a novel computationally efficient method called Hele-Shaw (HS) approximation is used for modeling fluid flow in FORGE well. An analysis of the influence of fracture intensity in a network is carried out using the HS method. The HS method was validated by solving the full Navier–Stokes equations (NSE) for a network of eight fractures. A good agreement was observed between the evaluated results (average deviation of 0.76%).

Large-time behavior of the solutions for an inflow problem of the full compressible Navier–Stokes–Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier–Stokes–Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L2-energy method, some time-decay estimates in Lp-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier–Stokes–Korteweg equations.

Global strong solutions with large oscillations to the 3D full compressible Navier–Stokes equations without heat conductivity

Global strong solutions with large oscillations to the 3D full compressible Navier–Stokes equations without heat conductivity

Global convergence rates from relaxed Euler equations to Navier–Stokes equations with Oldroyd-type constitutive laws

In a previous work (Peng and Zhao 2022 J. Math. Fluid Mech. 24 29), it is proved that the 1D full compressible Navier–Stokes equations for a Newtonian fluid can be approximated globally-in-time by a relaxed Euler-type system with Oldroyd’s derivatives and a revised Cattaneo’s constitutive law. These two relaxations turn the whole system into a first-order quasilinear hyperbolic one with partial dissipation. In this paper, we establish the global convergence rates between the smooth solutions to the relaxed Euler-type system and the Navier–Stokes equations over periodic domains. For this purpose, we use stream function techniques together with energy estimates for error systems. These techniques may be applicable to more complicated systems.

Stability of viscous contact wave for the full compressible Navier–Stokes–Korteweg equations with large perturbation

This study focuses on the stability of the viscous contact wave for the one-dimensional full Navier–Stokes–Korteweg equations with density-temperature dependent transport coefficients and large perturbation. Our findings demonstrate a Nishida–Smoller type result, indicating that the solution remains stable under large perturbation as long as γ−1 is sufficiently small. Notably, the smallness of the capillary coefficient is unnecessary. We then employ the initial layer analysis technique to investigate the asymptotic behaviour in the Wk,p norm. We show that the capillary term has a smoothing effect, which implies that the strong solution is indeed a smooth one. Our results represent an improvement over those previously reported in Chen and Sheng (2019 Nonlinearity 32 395–444). Furthermore, by applying the method in this study to the isothermal case, we can achieve a better outcome than Germain and LeFloch (2016 Commun. Pure Appl. Math. 69 3–61).

Aerodynamic and aeroacoustic design of electric ducted fans

This paper describes the aerodynamic and aeroacoustic design of an electric ducted fan (EDF). A prototype EDF was designed and built with three different rotor-stator sets to operate at design flow coefficients ϕd=[0.60,0.75,0.90]. Computational Fluid Dynamics (CFD) simulations show that changing the design flow coefficient changes the proportions of endwall and blade profile loss, with the ϕd=0.75 design providing the optimum balance. CFD predictions of aerodynamic performance are compared with experimental measurements and show good agreement. Data collected from full annulus Unsteady Reynolds-Averaged Navier Stokes (URANS) CFD simulations is used to predict tonal noise radiation using a numerical solution to Farassat's Formulation 1A, and compare favourably to acoustic measurements in an anechoic chamber. All three EDF iterations show similar levels of broadband noise radiation, although tonal noise radiation decreases with increasing design flow coefficient and results in more favourable Sound Quality Metrics performance. Design flow coefficient is shown to have a first-order impact on both the aerodynamic and aeroacoustic behaviour of the EDF, and it is proposed that an optimal design, encompassing both aerodynamic performance and reduced acoustic impact, can be achieved at design flow coefficients between ϕd=0.75 and ϕd=0.90.

CFD simulation study and experimental analysis of indoor air stratification in an unventilated classroom: A case study in Spain

Health problems and respiratory diseases are associated with poor indoor air ventilation. We investigated the air quality inside a classroom-laboratory where no ventilation is provided. The case of study, consisting of an internal enclosure, is located at the Escuela Técnica Superior de Edificación (ETSEM) of Madrid (Spain). The high height favours air stratification which is analysed in terms of temperature and CO2 spatial distribution. Temperature, air humidity, atmospheric pressure and CO2 concentration measurements were taken in time at three different height locations. A CFD numerical model was established to analyse air quality. Flow circulation is derived by solving full 3D Navier – Stokes governing equations, coupled with the thermal problem. The diffusion problem of the CO2 produced by the inner occupants is then derived from the kinematics solution. Three scenarios were taken into account: occupants seated (1), standing (2), half seated, half standing (3). Results clearly show the air stratification as a result of density gradient, which is in turn determined by temperature difference between the occupants and the surrounding air. Temperature prediction maximum relative error is contained to 3.5 %. As expected, CO2 concentration increases over time, reaching maximum values depending on the configuration considered and height location.

Asymptotic predictions on the velocity gradient statistics in low-Reynolds-number random flows: onset of skewness, intermittency and alignments

Stirring a fluid through a Gaussian forcing at a vanishingly small Reynolds number produces a Gaussian random field, while flows at higher Reynolds numbers exhibit non-Gaussianity, cascades, anomalous scaling and preferential alignments. Recent works (Yakhot & Donzis, Phys. Rev. Lett., vol. 119, 2017, 044501; Khurshid et al., Phys. Rev. E, vol. 107, 2023, 045102) investigated the onset of these turbulent hallmarks in low-Reynolds-number flows by focusing on the scaling of the velocity gradients and velocity increments. They showed that the onset of power-law scalings in the velocity gradient statistics occurs at low Reynolds numbers, with the scaling exponents being surprisingly similar to those in the inertial range of fully developed turbulence. In this work we address the onset of turbulent signatures in low-Reynolds-number flows from the viewpoint of the velocity gradient dynamics, giving insights into its rich statistical geometry. We combine a perturbation theory of the full Navier–Stokes equations with velocity gradient modelling. This procedure results in a stochastic model for the velocity gradient in which the model coefficients follow directly from the Navier–Stokes equations and statistical homogeneity constraints. The Fokker–Planck equation associated with our stochastic model admits an analytic solution that shows the onset of turbulent hallmarks at low Reynolds numbers: skewness, intermittency and preferential alignments arise in the velocity gradient statistics through a smooth transition as the Reynolds number increases. The model predictions are in excellent agreement with direct numerical simulations of low-Reynolds-number flows.

Simulation and validation of instability transient process in multistage high-speed axial compressor based on the body-force model

In this paper, a method for simulating the instability transient process of the axial compression system based on the body-force model is developed, and a corresponding simulation program is developed. Simulations of the transient process of instability were carried out on a high-speed four-stage compressor and compared with experimental data. At 50% of the design rotational speed, the type of instability was rotating stall, and the simulated and experimental stall cell propagation speed were very close to each other. At 70% of the design rotational speed, the type of instability was surge. A “surge loop” was simulated, and the surge period and the percentage of time spent in each phase were consistent with the experiments. The simulation successfully predicted the blockage in the surge re-pressurization phase, proving the reliability of the simulation results. In addition, the computation yields more information about the flow field. By summing the blade forces of all grids on a blade row by volume, the surge loadings are obtained. The analysis of the axial momentum equation shows that the obtained blade force variations are reasonable. The simulation time of the multistage axial compressor is greatly reduced compared to the full annulus three-dimensional unsteady Reynolds-averaged Navier–Stokes method, demonstrating its great advantage in the design phase of the compressor.

Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations

Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations

Vortex shedding from a square cylinder interacting with an undular bore wave train

In this work, we studied the wave-induced vortex generation and shedding from a semi-submerged vertical square cylinder interacting with an upcoming undular bore wave train under a shallow water configuration. This unsteady and rapid process was investigated by means of numerical and experimental approaches. A numerical simulation, solving the full turbulent viscous Navier–Stokes equations, was carried out in order to study and characterize both the undular bore wave properties and the vortex dynamics triggered during this unsteady interaction. Starting with fluid at rest, the undular bore was generated by the impulsive translational motion of a piston wavemaker at laboratory scale in both a numerical and an experimental wave tank. When the undular bore impinges on the cylinder, filamentary vortex structures were formed at the four cylinder's edges synchronized with the propagating wave motion, leading to the vortex shedding phenomena at a frequency that matched the wave instantaneous frequency. These vortices extended along the entire cylinder span under the water column, from the free surface to the seabed. At the trailing edge of the cylinder, a pairing process of two shed vortices was observed, similar to a Lamb–Oseen vortex pair. These vortices were present during the whole undular bore wave train dynamic forcing. An overall agreement was found with the experimental version of the bore–cylinder interaction, carried out in a physical wave tank. Laser sheet bore profiling and particle image velocimetry measurements of the velocity field confirmed undular bore properties, the onset of vortex formation, subsequent shedding, and pairing in the experiments performed in similar conditions with the numerical approach.

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

Corrections to long wavelength approximation of peristalsis viscous fluid flow within a wavy channel

This research paper delves into the pumping phenomenon of peristaltic motion within a wavy, straight channel. Our aim is to provide analytical solutions for the well-established long wavelength and small Reynolds number approximation. To achieve this, we derive third-order corrections to the ideal leading-order solutions. These closed-form solutions encompass the axial and transverse velocity components, pressure rise and axial pressure gradients, and drag force along the wavy channel. Additionally, we elucidate the trapping phenomena of viscous flowing boluses through stream function contours. Notably, these exact solutions enable the analysis of peristalsis with finite wave numbers and moderate Reynolds numbers, eliminating the need for numerical simulations of the full Navier–Stokes equations at such physical parameters. Our simulations reveal that the pumping region is significantly influenced by short wavelength and moderate Reynolds number corrections. Further, the corrections enhance the drag force, and cell divisions within the boluses are observed at higher Reynolds numbers. The analytical derivations provided save researchers from the need for full simulations of the complex equations to acquire the key insights and contributions to capture the essence of the physical peristalsis mechanism inside wavy channels.

Stable production of fluid jets with vanishing diameters via tip streaming

We study numerically the microjetting mode obtained when a fluid is injected through a tube submerged in a uniaxial extensional flow. The steady solution to the full nonlinear Navier–Stokes equations is calculated. We obtain the linear global modes determining the linear stability of the steady solution. For sufficiently large outer viscosity, the flow remains stable for infinitely small values of the injected flow rate. This implies that jets with vanishing diameters can be produced regardless of the jet viscosity and outer flow strength. For a sufficiently small inner-to-outer viscosity ratio, the microjetting instability is associated only with the flow near the entrance of the jet. The tapering meniscus stretches and adopts a slender quasiconical shape. Consequently, the cone tip is exposed to an intense outer flow, which stabilizes the flow in the cone–jet transition region. This work presents the first evidence that fluid jets with arbitrarily small diameters can be stably produced via tip streaming. The results are related to those of a droplet in a uniaxial extensional flow with its equator pinned to an infinitely thin ring. The pinning of the equator drastically affects the droplet stability and breakup.

A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations

AbstractWe propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low‐order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two‐step technique: computation of a tentative flux, here, with a centered approximation of the velocity, and limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation (with the velocity, one of its component, the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi‐implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi‐implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.

Analytical solution for laminar entrance flow in circular pipes

This study introduces an analytical solution for the laminar entrance flow in circular pipes, aiming to confirm the occurrence of velocity overshoot. Velocity overshoot is characterised by the maximum axial velocity appearing near the pipe wall instead of the central axis. Similar to the previous studies, the analytical solution is derived from the parabolised Navier–Stokes equation; however, the specific approach used in linearising the momentum equation has not been attempted before. The accuracy of this analytical solution has been verified through a comprehensive comparison with various published experimental data. The existence of velocity overshoot at a short distance from the inlet, which is evident in numerous numerical calculations based on the full Navier–Stokes equations and corroborated by recent magnetic resonance (MR) velocimetry experiments, is identified analytically for the first time. The parabolised Navier–Stokes equation has inherent self-similarity with respect to the Reynolds number, implying that $Re$ is incorporated into the dimensionless variables rather than serving as an independent flow parameter. According to both MR velocimetry measurements and the present analytical solution, the self-similarity is not valid immediately following the pipe inlet, and this becomes more evident as $Re$ decreases; hence, the analytical solution derived from the parabolised Navier–Stokes equation cannot accurately predict the evolution of the velocity profile within this region near the pipe inlet.

Energy equality of weak solutions of the Navier–Stokes–Fourier equations allowing vacuum

In this paper, we are concerned with the conservation of energy criterion of the weak solutions to the Navier–Stokes–Fourier equations on the torus allowing vacuum. We establish the energy conservation criteria via the gradient of velocity for the weak solutions of this system, which generalizes the corresponding recent energy conservation criteria in terms of the velocity obtained by Aoki and Iwabuchi in Aoki and Iwabuchi (2022). Moreover, sufficient conditions for weak solutions keeping the energy balance allowing vacuum are shown for the full Navier–Stokes equations.

A numerical investigation of momentum flux and kinetic energy transfers between turbulent wind and propagating waves

This paper focuses on simulating turbulent flow over propagating waves by solving the full Navier–Stokes equations in a moving frame. A careful comparison of flow statistics with previous experimental and numerical results demonstrates, to some extent, the rationality of simplifying wind waves as turbulent flow over moving wave boundaries. The phase-averaging method is then applied to investigate the momentum and energy transfers between turbulent wind and waves propagating at slow, intermediate and fast speeds. The results suggest that the dominant mechanism for producing Reynolds shear stress (RSS) and turbulent kinetic energy (TKE) is related to the wave age. Slow waves produce RSS and TKE similar to a two-dimensional shear turbulence. However, a fast wave enhances the streamwise Reynolds normal stress, the windward side's negative RSS and the gradient of both streamwise and vertical velocities, leading to additional RSS and TKE productions that can be ignored under the slow wave regimes. A strengthening wave–turbulence exchange is also found for fast waves. The intermediate wave can be regarded as a transitional condition determining this change.