Steady Navier-Stokes Equations Research Articles

Performance improvement of a rotating gas separator for electrical submersible pump considering entropy generation method

The utilization of Electrical Submersible Pumps (ESPs) stands as one of the foremost methods in artificially extracting oil, primarily within the oil industry. Under typical conditions, the pump encounters a two-phase flow of oil and gas. Should the gas phase within this mixture surpass an acceptable threshold, it can lead to a significant decline in performance and inflict detrimental damage upon the pump. To address this issue, the incorporation of a gas separator unit, one of the pivotal components within ESPs, is imperative at the pump's inlet section. Enhancing the gas separator unit holds paramount importance for ensuring pump safety and optimizing performance, particularly in wells with elevated gas contents.The conventional approach to calculating hydraulic losses in turbomachinery, relying on pressure drop calculations, often falls short in pinpointing the precise locations of losses. This paper delves into enhancing gas separator performance by augmenting gas separation efficiency through the application of the entropy generation method and leveraging the second law of thermodynamics. To achieve this, pivotal entropy-generating geometrical parameters of the gas separator were identified as the design variables necessary for forming the Taguchi sample space. Numerical simulations were conducted utilizing steady three-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations for incompressible fluid flow, coupled with a turbulence model. Results indicate that the gas separation efficiency (GSE) of the initial model surged from 45.6% to 88% in the improved sample, albeit with a concurrent rise in entropy generation rate (EGR) from 209 W/K to 243.1 W/K. To establish a more comprehensive criterion for analysis, the ratio of GSE to EGR was employed as an objective function. This ratio quantifies the efficiency of gas separation per unit of energy loss, and it escalated from 0.218 in the primary sample to 0.362 in the improved sample, representing a 66% increase. This signifies that despite simultaneous increases in EGR and GSE, the rate of energy loss per unit of gas separation has diminished. Furthermore, the heightened EGR in the improved sample can be attributed to an increase in total volume, leading to an expanded contact surface area and consequently an elevation in total entropy. This investigation underscores the advantages of the entropy generation method, particularly in delineating the precise location and magnitude of energy dissipation within the rotating gas separator.

On Leray’s problem in an infinitely long pipe with the Navier-slip boundary condition

The original Leray's problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach to the Poiseuille flow subject to the no-slip boundary condition at spacial infinity. In this paper, the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition, is addressed. Due to the complexity of the boundary condition, some new ideas, presented as follows, are introduced to handle the extra difficulties caused by boundary terms. First, the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced, which will be served as the asymptotic profile of the solution to the generalized Leray's problem at spacial infinity. Second, a solenoidal vector function defined in the whole pipe, satisfying the Navier-slip boundary condition, having the designated flux and equalling to the Poiseuille flow at large distance, will be carefully constructed. This plays an important role in reformulating our problem. Third, the energy estimates depend on a combined $L^2$ estimate of the gradient and the stress tensor of the velocity.

Numerical Study of the Effect of the Ejection Angle on the Cooling Performance of a High-Pressure Gas Turbine

In modern gas turbines, the purge flow plays an important role in improving the performance of the gas turbine, as it is adapted against the problem of the ingestion of hot gases in the inter-disc cavity of the turbine. The performance of this purge flow is necessarily influenced by some geometrical and physical parameters. This paper presents a 3D numerical study on the influence of the ejection angle on the cooling performance in a high pressure transonic gas turbine with the presence of the cavity. Five inclination cavity angles are compared (𝝓 = 30°, 45°,60°,75°and 90°) by using ANSYS Fluent, a steady flow Reynolds average Navier-Stokes equations are solved, and the turbulence model k-ω SST has been chosen based on comparison of the distribution of velocity and heat transfer convective coefficient with experimental results in a previous work. It was seen that that cooling is limited to the suction side because of the non-uniform pressure distribution of the main flow and the high pressure of the stagnation point. The optimal configuration was determined for an ejection angle of 30°.

The scalar Oseen problem in two-dimensional exterior domains

In the present paper, we investigate the scalar Oseen equation in a two-dimensional exterior domain Ω, with Dirichlet boundary condition. This problem represents a linearized form of the steady Navier-Stokes equations. In order to better control the behavior at infinity of functions, the use of weighted Sobolev spaces as a functional framework turns out to be adequate. The established results are related to the existence and the uniqueness of weak solutions on Lp-theory for any real 1<p<∞. Our analysis is mainly based on the principle that linear exterior problems can be solved by combining their properties in bounded domains which are also established in this paper and their properties well known in the whole space R2.

Heat transfer enhancement of supercritical carbon dioxide in eccentrical helical tubes

A type of eccentrical helical tube (EHT) was proposed for more efficient and compact supercritical carbon dioxide (SCO2) heat exchangers in refrigeration systems. The forced convective heat transfer of SCO2 flowing in three designed eccentrical helical tubes with pitches of 25, 50 mm and eccentricities of 0.9, 1.5 mm was simulated by using the three-dimensional steady Reynolds-averaged Navier-Stokes equations, shear stress transport turbulence model and energy equation in ANSYS CFX 2019 R2 at mass fluxes of 200, 400 kg/m2s, inlet pressures of 8, 9 MPa, outwards wall heat fluxes of 12, 24 kW/m2. The flow and heat transfer models were validated in the plain tube with experimental mean heat transfer coefficient and empirical Darcy friction factor. Influences of pitch, eccentricity and operational conditions on heat transfer enhancement were identified. Flow pattern, heat transfer mechanism and vortex kinematics were clarified. Heat transfer enhancement of SCO2 in the EHT was compared with water and air in the EHT, twisted elliptical tube and conical tube, respectively. A pitch of 25 mm and eccentricity of 1.5 mm at 6 mm tube diameter can achieve a better thermal-hydraulic performance. A higher SCO2 mass flux raises Nusselt number and reduces friction factor, a higher inlet pressure and a larger wall heat flux reduces Nusselt number but increases friction factor. The ratio f/f0 is ranged in 2.52–1.68 and drops off with increasing inlet pressure and nominal Reynolds number, the ratios Nu/Nu0, ψ and η curves show a concave shape near the pseudocritical point ranged in 1.68–1.06 and 1.06–1.55, 1.23–0.83 and 0.83–1.30, 0.67–0.50 and 0.50–0.92 at the pitch of 25 mm and eccentricity of 1.5 mm, inlet pressure of 8 MPa, mass flux of 400 kg/m2s, and wall heat flux of 12 kW/m2. A helical flow pattern with a core flow occurs in the EHT, the local heat transfer coefficient can be correlated to absolute helicity. The high-wall shear stress on the ridge of the EHT is responsible for the heat transfer enhancement. The f/f0 value of the EHT is in between the twisted elliptical tube and the conical tube, but the Nu/Nu0 and ψ values are comparable.

On the multi-objective perspective of discrete topology optimization in fluid-structure interaction problems

Fluid-structure interaction is a challenging topic that addresses fluid and solid physics, as well as the stress coupling between them. Traditional topology optimization methods are performed with coupling load interpolation schemes in order to have some information on fluid flow sensitivity analysis and improve compliance minimization into design-dependent problems. In this paper, we propose a strategy to design fluid-structure systems by combining solid and fluid objectives without the need for the coupling load interpolation scheme. Therefore, we investigate topology optimization applied to fluid-structure interaction problems via a multi-objective formulation. We combine structural compliance with some fluid flow objective functions (energy dissipation, downforce, and drag) subject to a volume constraint. We assume linear elasticity for the solid and the steady incompressible Navier-Stokes equations for the fluid. The optimization problem is solved by using sequential integer linear programming via the TOBS (Topology Optimization of Binary Structures) method with a geometry trimming (GT) technique. It is a gradient-based method that produces explicit (discrete) boundaries that are convenient for coupled physics problems. Numerical results are presented for two and three-dimensional problems considering low to moderate Reynolds numbers. We demonstrate that the proposed multi-objective approach yields physically meaningful designs with improved performance, highlighting that the incorporation of coupling load interpolation into fluid-structure interaction optimization is redundant.

Uniqueness of steady Navier-Stokes under large data by continuous data assimilation

We propose a continuous data assimilation (CDA) method to address the uniqueness problem for steady Navier-Stokes equations (NSE). The CDA method incorporates spatial observations into the NSE, and we prove that with sufficient observations, the CDA-NSE system is well-posed even for large data where multiple solutions may exist. This CDA idea is in general helpful to determine solution for non-uniqueness partial differential equations (PDEs).

Numerical Modeling of the Flow around a Cylinder using FEATool Multiphysics

The current study examines the numerical analysis of the laminar flow around a cylinder at various Reynolds numbers (0.1, 1.1, 20, 26, 50, 100, and 195). The research found that a steady state can exist for Reynolds number values of 0.1, 1.1, 20, and 26. However, the flow pattern becomes unstable at Reynolds numbers 50, 100, and 195, leading to the development of the Kármán vortex street. The FEATool Multiphysics software in MATLAB (R2019b) was utilized to numerically solve the steady 2D Navier-Stokes equation. The study compared the estimated drag coefficient to previous experimental and analytical studies in Abaqus/CFD. The lift and pressure coefficients were also calculated, and their results were found to be in strong agreement with earlier investigations in terms of predicting pressure and velocity distribution. The analysis provided insight into how the flow field changes with increasing Reynolds numbers.

Impact of eave and roof pitch on cross ventilation for an isolated building with sawtooth roof

An eave refers to an extension attached to the building roof to protect the interior space from direct solar radiation and improve the performance on cross ventilation. In this study, the impact of eave inclination angle and roof pitch of an isolated sawtooth roof building on cross ventilation were investigated. The eave configurations at either windward or leeward openings were included. 3D steady Reynolds-Averaged Navier-Stokes (RANS) equation in combination with the Shear-Stress Transport model (SST k-ω model) was used for the Computational Fluid Dynamics (CFD) simulations. Grid sensitivity study was carried out and the performance of cross ventilation was evaluated based on the non-dimensional velocity magnitude, spatial distribution of pressure coefficient as well as the ventilation rate of the building. For the simulation model with 55° roof pitch, it is observed that a region with high velocity magnitude formed on top of the leeward eave due to the higher roof pitch and presence of the leeward eave. Results also indicated that the building model with 90° leeward eave and 55° roof pitch has the highest increment in ventilation rate which is 7.16%. On the other hand, the building model with 90° windward eave has the highest pressure coefficient because more blockage of airflow is caused by a steeper roof as the roof pitch of the building increases. Furthermore, the building model with 90° leeward eave shows a larger region with negative pressure at the leeward façade indicating higher airflow leaving the leeward opening. Therefore, the airflow behavior and characteristic are both dependent on the roof pitch and eave inclination angle for a naturally ventilated building.

Optimizing the trench shaped film cooling design

In this numerical study an optimized trench film cooling design is determined using a Bayesian algorithm and neural network trained RANS model. Three objective functions were considered, the area-averaged film cooling efficiency, spatial standard deviation of film cooling efficiency and hot gas ingestion into the trench. Nine geometrical design parameters were varied to allow for a 3D trench shape and to find an optimal trench design based on the initial parametrization of the trench. Jet-engine like inflow boundary conditions with respect to turbulence intensity and length scales were applied. The investigated momentum ratios (I) were 1 and 8 at a main flow Reynolds number (ReD) of 2500. For each design the steady state Reynolds Averaged Navier-Stokes (RANS) equations were solved using the commercial Computational Fluid Dynamics (CFD) code Ansys Fluent V2022 R1. The turbulence model coefficients of the generalized k−ω (GEKO) model were tuned to approximate the time-averaged 3D temperature field from a predictive Large Eddy Simulation (LES) and trained by a neural network to improve the prediction capability. The tuned GEKO model shows improved agreement with experimental data of a literature case compared to the standard GEKO model. With this tuned RANS model optimized trench designs are found and validated by additional LES’s. The optimized designs include angled side walls to improve former trench designs, particularly in mitigating hot gas entrainment into the trench, which could be omitted almost entirely.

Parametric optimization of vortex generator configuration for flow control in an intake duct for waterjet propulsion

Vortex generators (VG) are simple miniature devices that suppress flow separation in aeronautical elements. In this study, VG device operation in a submerged flush waterjet inlet was numerically investigated to improve the flow quality entering the jet pump at low Inlet Velocity Ratio (IVR). Steady Reynolds-averaged Navier-Stokes equations are solved using the SST k-ω turbulence model and used to simulate the flow field, with the effects of the orientation, angle of attack, height and installation position of VGs thoroughly analyzed. The hydrodynamic performance of the intake duct was assessed over the operational range. The results show that VGs arranged in a divergent configuration better suppress flow separation. The zonal magnitude of the high turbulent kinetic energy on the ramp side was reduced, with the non-uniformity in the pump face plane reduced from 0.41 to 0.26. Excessive angles of attack and VG heights have negative effects. Through parametric studies, VGs were found to have best performance in the target duct with an angle of attack of 40⁰ and a height measuring 20% of the bottom boundary layer thickness. In addition, VGs installed along the keel of a ship can effectively suppress the flow distortion and improve the overall efficiency of the intake duct.

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in R3. We assume that the flow is periodic in x3-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. This reveals the relation between the integrability of ∇v and the decay rate of ω near the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

Convective heat transfer in a thermal chimney for freshwater production in geothermal total flow systems

The convective heat transfer of air in a laboratory-scale thermal chimney with rectangular cross-section of constant area and two row electrical heaters simulating two heat exchangers was studied experimentally and numerically at 60–200 ℃ nominal temperatures of the top row heaters, 100 ℃ of the low row heaters and 20 ℃ ambient temperature to verify our design concept on freshwater production in geothermal total flow systems. Computational fluid dynamics simulations of air convective heat transfer were performed in ANSYS 2019R CFX based on the three-dimensional, steady Reynolds-averaged Navier-Stokes equations, Boussinesq buoyancy model, k-ω turbulence model, and energy equation. The thermal radiation between heater surfaces and chimney walls was considered. The overall thermal and heat transfer characteristics, temperature and flow fields in the chimney were obtained. Effects of boundary condition of heater surface and thermal radiation between two row heaters on heat transfer were discussed. The thermal characteristics of the chimney with two row heaters are better than that with single row heaters. The predicted thermal power and convective Nusselt number agree with the experimental data, and the convective Nusselt number of the low row heaters is enhanced by (11.6–29.8) % compared with the single row heaters. The optimal operating nominal temperature of top row heaters should be higher than 140 ℃, and the optimal centre-to-centre row gap ratio is 5. Multiple jets in the gaps among the heaters and temperature jump crossing each row were observed. The maximum velocity and temperature jump rise with increasing heater nominal temperature.

A theoretical model for calculating particle trajectory in a vane-type separator

A vane-type separator has been successfully applied to gas-liquid and liquid-liquid separation. In order to evaluate the separation performance of the separator, a theoretical model for calculating the particle motion trajectory in the vane-type separator was developed. Firstly, the simplified steady incompressible Navier-Stokes equations were derived to acquire the equations of tangential velocity and axial velocity of continuous phase flow field. Then, the coefficients were determined by fitting the CFD results and it can be used to obtain the velocity at different swirler structure conditions. Further, the force analysis on the particle was conducted to develop the motion governing equations to predict the trajectory. The predicted separation lengths at different geometric swirl number conditions show a good agreement with the experimental results and its relative error is smaller than 3%. Besides, the effects of the key parameters including density difference, particle size and viscosity of the continuous phase on the separation length were investigated.

Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

We investigate the problem of classification of solutions for the steady Navier–Stokes equations in any cone-like domain. In the form of separated variables, u(x,y)=φ1(r)v1(θ)φ2(r)v2(θ), where and in the polar coordinates, we obtain the expressions of all smooth solutions with C 0 Dirichlet boundary condition. In particular, we find some solutions which are Hölder continuous on the boundary but their gradients blow up at the corner, show that all solutions in the entire plane must be polynomials, and prove a sharp Liouville type theorem.

On the steady Prandtl boundary layer expansions

In this paper, we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in (0,L) × ℝ+ with no-slip boundary conditions. By estimating the stream-function of the remainder, we justify the validity of the Prandtl boundary layer expansions. Specially, we show the global stability under the concavity condition of the Prandtl profile for an arbitrarily large constant L when the Euler flow is shear.

Existence of solutions to steady Navier-Stokes equations via a minimax approach

Our objective in this paper is to develop and utilize a minimax principle for proving the existence of symmetric solutions for the stationary Navier-Stokes equations. Notwithstanding its application to symmetric solutions in this paper, our minimax principle is broad enough to capture other types of solutions provided the equation and the external force are compatible under a family of operations including but not limited to being invariant by compact groups. The subset of functions compatible under this family of operations is not required to be a linear subspace, and being a closed convex set suffices for our purpose.

Superlinear convergence of Anderson accelerated Newton's method for solving stationary Navier–Stokes equations

AbstractThis article studies the performance Newton's iteration applied with Anderson acceleration for solving the incompressible steady Navier–Stokes equations. We manifest that this method converges superlinearly with a good initial guess, and moreover, a large Anderson depth decelerates the convergence speed comparing to a small Anderson depth. We observe that the numerical tests confirm these analytical convergence results, and in addition, Anderson acceleration sometimes enlarges the domain of convergence for Newton's method.

Differentiability Properties for Boundary Control of Fluid-Structure Interactions of Linear Elasticity with Navier-Stokes Equations with Mixed-Boundary Conditions in a Channel

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

Improved convergence of the Arrow–Hurwicz iteration for the Navier–Stokes equation via grad–div stabilization and Anderson acceleration

We consider two modifications of the Arrow–Hurwicz (AH) iteration for solving the incompressible steady Navier–Stokes equations for the purpose of accelerating the algorithm: grad–div stabilization, and Anderson acceleration. AH is a classical iteration for general saddle point linear systems and it was later extended to Navier–Stokes iterations in the 1970’s which has recently come under study again. We apply recently developed ideas for grad–div stabilization and divergence-free finite element methods along with Anderson acceleration of fixed point iterations to AH in order to improve its convergence. Analytical and numerical results show that each of these methods improves AH convergence, but the combination of them yields an efficient and effective method that is competitive with more commonly used solvers.