Stationary Flow Research Articles

Exact solutions of a hybrid nanofluid model for the flow across a heated stretching cylinder at a stagnation point

We examined the two-dimensional, Stagnation flow across a thermally extending cylinder and identify the precise analytical solutions. As hybrid nanoparticles, Copper (Cu) and Alumina Oxide solid [Formula: see text] particles are employed with base fluid [Formula: see text]. We added nanoparticles of Copper (Cu) and Alumina Oxide solid [Formula: see text] to the water being that its heat permeability is low. The fundamental fluid’s ([Formula: see text]) thermal efficiency has risen as a result. A computational model of stationary point flow through a heated stretching cylinder is constructed using the theory behind the mechanism of the flow strategy. Carefully considered is the interaction between the impinging stagnation flow and the flow produced by stretching the cylinder’s surface. For this fascinating subject, precise analytical solutions are obtained, and the results are also graphically evaluated. The flow picture is evaluated using streams for any indications of evolved flow stretching or impinging stagnation flow dominating under pressure. A thorough investigation of heat transport is also offered, along with a graphical description of the Nusselt number (Nu). When the stretching cylinder has a lower stretching velocity A, an impinging stagnation flow that is more forceful or completely developed is seen for larger values of the approaching flow pressure parameter. When greater approaching pressure P flow values are applied, stagnation flow dominates over stretching flow. A significant portion of stagnation point flow research takes a numerical approach to the subject; nevertheless, in this work, we seek to present a precise solution to the challenge at hand. Behavior of fluid as stream lines and graphs of velocity profile, temperature profiles are plotted by using Mathematica software.

Research on the flow characteristics of injecting hydrogen into natural gas pipelines and its impact on energy metering

Injecting hydrogen into natural gas pipelines for direct use by end users is an economical and effective method to realize long-distance transportation of hydrogen. However, the density of hydrogen is much lower than natural gas, and it tends to accumulate at the top of the pipeline, resulting in significant different flow characteristics of pure natural gas. Therefore, mixing hydrogen into natural gas pipelines will affect the accuracy of calorific value and flow rate measurement in the existing natural gas metering system. This article studies the flow characteristics of injecting hydrogen into natural gas using a T-shaped pipe and its impact on energy measurement through CFD numerical simulation. The results show that the energy metering equipment should be installed where the gas reaches a uniform mixture and the radial section velocity is centrally symmetrical. Injecting hydrogen with a lower position, and smaller main stream velocity will shorten the distance between the metering pipeline section and the mixing point, but a distance of at least 100 times the pipe diameter is still required. Thus, a mixing process combining a static mixer and flow straightener is proposed, which reduces this distance to within 40 times the pipe diameters.

Early-stage Development of the CoRISMA Mechanical Circulatory Support (CMCS) System for Heart Failure Therapy.

CoRISMA MCS Systems Inc (Hamden CT) is developing an innovative mechanical circulatory support system (CMCS) as a durable therapeutic option for heart failure (HF) patients. The CMCS system is comprised of an axial flow pump, non-contacting hydrodynamic bearings, and integrated DC motor designed to be fully implantable in a left atrial (LA) to aortic (Ao) configuration; this unloading strategy may be particularly beneficial for HF patients with preserved ejection fraction (HFpEF). The small (5.5 cm3), lightweight (20g), and low power (5-7W) device design should allow for a less invasive off-pump implant. We present early-stage engineering development and testing of the prototype CoRISMA pumps. Computational fluid dynamics (CFD) modeling was performed to evaluate flow and shear in two impeller (3 blades, 0.5mm thickness, 8.9mm diameter, 0.15mm gap, polished titanium) and diffusor (5 blades, polished titanium) candidate designs. Test apparatuses were custom built to expedite development of the impeller/diffuser designs and iteratively refine the CFD models. Two candidate impeller/diffusor designs were fabricated and tested in each of the two test apparatuses (n = 4 impeller/diffuser + test fixture configurations) in static mock flow loops (hydrodynamic H-Q curves, 3.5 cP glycerol solution at 37°C), and in dynamic mock flow loops (hemodynamics, 3.5 cP glycerol solution at 37°C) tuned to HF conditions (mean aortic pressure 50 mmHg, central venous pressure 15 mmHg, aortic flow 3.0L/min, and heart rate 80bpm). CFD predicted flows of 4.56L/min and 4.82L/min at 100 mmHg for impellers/diffusers 1 and 2, respectively. Impeller 2 required less torque to generate a 6% increase in fluidic flow, and the diffuser had a larger area of high pressure, indicative of lower friction, which likely contributed to the increased efficiency. Experimental testing for all four configurations in the static and dynamic mock loops met performance metrics as evidenced by generating 4.0-4.5L/min flow against 70-76 mmHg pressure at 25,000rpm and restoring hemodynamics in the dynamic mock flow loop (MAP = 80 mmHg, CVP = 0 mmHg, total flow = 5.5L/min) from baseline simulated HF test conditions. These results demonstrate proof-of-concept of the early engineering design and performance of the prototype CoRISMA pumps. Engineering specifications, challenges observed, and proposed solutions for the next design iteration were identified for the continued development of an effective, reliable, and safe LA-to-Ao CMCS system for HF patients. Current design plans are underway for incorporating a wireless energy transfer system for communication and power, eliminating the need for and complications associated with an external driveline, to achieve a fully-implantable system.

Electrochemical treatment of azo dye wastewater by dynamic flow diaphragm system: Response surface methodology optimization and energy consumption analysis

In this experiment, the degradation process of azo dye wastewater was studied based on the diaphragm double chamber electrochemical reactor, taking the mono azo dye amaranth as an example. Through the circulation flow of appropriate volume of wastewater in the cathode and anode chamber, the simultaneous action of cathode and anode was realized, and the degradation and decolorization efficiency of azo wastewater was improved while reducing energy waste. The optimal reaction conditions optimized by response surface methodology were as follows: the voltage was 20.74 V, the plate spacing was 4.89 cm, and the cycle speed was 60.66 rad / min. The optimal decolorization rate was 99.66 %. The energy consumption analysis of the dynamic flow diaphragm system electrochemical method and the static flow diaphragm system electrochemical method for the treatment of azo dye wastewater was compared and analyzed. In addition, the possible degradation process of azo dye wastewater by dynamic flow electrochemistry was inferred by ultraviolet spectroscopy and mechanism prediction analysis. The experimental results show the great potential of the dynamic flow membrane system process in the treatment of azo dyes.

Influence of Substrate Structure and Associated Properties on Endothelial Cell Behavior in the Context of Behaviors Associated with Laminar Flow Conditions.

In order to treat most vascular diseases, arterial grafts are commonly employed for replacing small-diameter vessels, yet they often cause thrombosis. The growth of endothelial cells along the interior surfaces of these grafts (substrates) is critical to mitigate thrombosis. Typically, endothelial cells are cultured inside these grafts under laminar flow conditions to emulate the native environment of blood vessels and produce an endothelium. Alternatively, the substrate structure could have a similar influence on endothelial cell behavior as laminar flow conditions. In this study, we investigated whether substrates with aligned fiber structures could induce responses in human umbilical vein endothelial cells (HUVECs) akin to those elicited by laminar flow. Our observations revealed that HUVECs on aligned substrates displayed significant morphological changes, aligning parallel to the fibers, similar to effects reported under laminar flow conditions. Conversely, HUVECs on random substrates maintained their characteristic cobblestone appearance. Notably, cell migration was more significant on aligned substrates. Also, we observed that while vWF expression was similar between both substrates, the HUVECs on aligned substrates showed more expression of platelet/endothelial cell adhesion molecule-1 (PECAM-1/CD31), laminin, and collagen IV. Additionally, these cells exhibited increased gene expression related to critical functions such as proliferation, extracellular matrix production, cytoskeletal reorganization, autophagy, and antithrombotic activity. These findings indicated that aligned substrates enhanced endothelial growth and behavior compared to random substrates. These improvements are similar to the beneficial effects of laminar flow on endothelial cells, which are well-documented compared to static or turbulent flow conditions.

Synthetic ester-based forced flow immersion cooling technique for fast discharging lithium-ion battery packs

Rapid advancements in Li-ion battery technology are being made to meet the growing demand for efficient energy storage solutions in electric vehicles and portable electronics. However, heat generation during rapid charging and discharging remains a significant challenge, as it can lead to overheating, fire, and explosion. To address this issue, battery thermal management systems require improvements in cooling strategies. This study aims to optimize the thermal performance of Li-ion battery packs during fast discharge operation by single-phase synthetic ester oil-based forced flow immersion cooling (FFIC) technique. The study analyzes the thermal performance of the FFIC of a 4S2P LIB pack based on various flow rates, pressure drops, and pump power consumption and compares it with various conventional cooling methods and dielectric fluids. The MSMD-based NTGK battery modeling approach is used to analyze the electrical and thermal characteristics of the battery pack at 3C discharge and environmental temperature of 25 °C. The simulation findings revealed that the synthetic ester oil has a superior cooling effect than mineral oil in the 4S2P Li-ion battery pack. The experimental results showed that the synthetic ester-based FFIC Li-ion battery pack achieves an optimal pack temperature of 31.3 °C and uniform temperature distribution at the flow rate of 5 L/min and pressure drop of 228.44 Pa. Moreover, FFIC reduces temperature rise by 51 % compared with natural air convection and 35 % compared with static flow immersion cooling (SFIC) in the 4S2P lithium-ion battery pack. Also, the practical significance of the SEO-based FFIC immersion cooling technique is assessed across extreme climate zones such as −10 °C and 40 °C and determined its superior performance in Li-ion batteries. Hence, this study proved that the FFIC technique is suitable for Li-ion batteries in electric vehicle applications in terms of optimal performance, reliability, and safety during high-current discharge operations.

Experimentally estimating wind load coefficients for tornadoes – An alternative perspective

Given the increased interest in tornado-induced wind loading (in part exhibited by inclusion of such loading in wind standards around the world) it is vital to understand the various characteristics of such loading and their relative overall importance. As such, employing a range of types of simulations and simulators would be helpful for understanding and estimating tornado-induced wind load coefficients. This paper advocates a quasi-steady framework to estimate tornado-induced wind loads and identifies the tornado flow characteristics most likely to influence these loads. The flow characteristics discussed include the flow field itself, the static pressure field, vortex translation, flow turbulence, streamline curvature, and flow acceleration. A fundamental conclusion of this paper is that pressure coefficients for tornado-induced wind loading should always be measured and reported as functions of these characteristics. The paper discusses various alternatives for simulating these characteristics and highlights which types of facilities would be effective for studying each one. Such approaches will also require deliberately measuring velocity and static pressure simultaneously with any load measurements made on a building model. This will then help clarify the dependencies of pressure and load coefficients to the various parameters, will limit the parameter space that must be explored to understand extreme loading, and will facilitate easier comparison among different laboratory results, all of which will ultimately lead to improved design standards.

A unified macroscopic equation for creeping, inertial, transitional, and turbulent fluid flows through porous media

Over the course of several decades, numerous model equations of the macroscopic fluid flow through porous media have been proposed. The application of such equations is, however, often complicated due to the requirement of variant specifications of parameters and empirical factors for different flow regimes. It is, therefore, necessary and desirable to have a unified fundamental equation that is capable of predicting porous media flows for the entire spectrum of flow regimes that are practically encountered. This work aims to fulfill that requirement. With the aid of a hypothesis-based analysis, finite-element simulations, and published experimental data, a new macroscopic transport equation has been proposed to predict statistically stationary single-phase incompressible flows through a non-deformable stationary porous medium. The new model may be written as a drag law associated with a dimensionless resistance parameter that is a function of the porous medium geometry and the flow forces. Though complex, this resistance parameter may be modeled as a power function in terms of three predictable parameters. Overall, the proposed transport equation has been found to be a more extensive form of other key models in existence. Using approximately 6000 analytical, numerical, and experimental data points, the equation has been validated as an excellent model for creeping, inertial, transitional, and turbulent porous media flows. The results show that the proposed equation is applicable to simple and complex porous media of 30%–90% porosity. Moreover, a dimensionless group in terms of the equation's resistance parameter has been established as useful for scaling.

Effect of secondary flow and wall collisions on particle-laden flows in 90° pipe bends

Fully developed, statistically stationary particle-laden turbulent flows in a 90° pipe bend at a moderate Reynolds number are studied using direct numerical simulation coupled to a Lagrangian particle tracking technique. Three populations of particles are studied which are two-way coupled with the carrier phase. The focus is the investigation of the effect of strong curvature on particle transport dynamics across a range of particle inertias. A validation of the carrier phase predictions is performed, with good agreement found with available experimental data. It is demonstrated that the presence of particles notably affects the turbulence statistics of the carrier phase, decreasing the mean fluid velocity in the outer bend while slightly increasing it in the inner bend. Two intriguing phenomena emerge: the presence of a particle void near the inner bend and reflection layers near the outer bend. Centrifugal forces are responsible for driving particles to the outer wall, with some rebounding back into the main body of the flow, and others, influenced by the secondary flow in the plane of the pipe cross-section, converging towards the inner wall before re-entering the central pipe region. These effects dictate the size and shape of the particle void region and the formation of reflection layers due to particle-wall collisions. These two phenomena have a significant influence on the particle distribution in the pipe bend, as well as on the first- and second-order moments of the velocity and acceleration of the particulate phase. Finally, a heat map of particle-wall collision statistics indicates that the effect of the secondary flow on particle distribution and particle-wall collisions is not negligible.

Исследование характеристик стационарных режимов движения подводной привязной системы «ТНПА – кабель – судно»

The efficiency of using uninhabited underwater vehicles of the tethered type is associated with the need for its propulsion and steering complex (PSC) to compensate for the reaction of the communication cable of the device with the support vessel when maneuvering near the work site. The work examines the achievable boundaries of the maneuvering zone of a remotely controlled uninhabited underwater vehicle (ROV) for known values of the length of the communication cable, the submersion depth of the device, the current speed in the work area and the traction characteristics of the PSC. Algorithms for determining the boundaries of the working area and the achievable speed of the apparatus are presented, based on calculating the tension at the ends of the communication cable in a stationary flow, using the equations of a catenary line and the numerical integration of the equations of an inextensible flexible thread. In order to increase the operational reliability of underwater technical work, recommendations have been developed for choosing a cable length that prevents it from sagging below the submersion depth of the apparatus with possible snagging and damage by bottom objects. During the model experiment, the boundaries of the maneuvering zone were assessed relative to the coordinates of the carrier for a promising ROV with known hydrodynamic resistance and traction characteristics of the PSC, as well as the dependence of the achievable speeds of the vehicle within the maneuvering zone on the immersion depth and the speed of the oncoming current.

Molecular dynamics study of the sonic horizon of microscopic Laval nozzles.

A Laval nozzle can accelerate expanding gas above supersonic velocities, while cooling the gas in the process. This work investigates this process for microscopic Laval nozzles by means of nonequilibrium molecular dynamics simulations of stationary flow, using grand-canonical Monte Carlo particle reservoirs. We study the steady-state expansion of a simple fluid, a monoatomic gas interacting via a Lennard-Jones potential, through an idealized nozzle with atomically smooth walls. We obtain the thermodynamic state variables pressure, density, and temperature but also the Knudsen number, speed of sound, velocity, and the corresponding Mach number of the expanding gas for nozzles of different sizes. We find that the temperature is well defined in the sense that the each velocity components of the particles obey the Maxwell-Boltzmann distribution, but it is anisotropic, especially for small nozzles. The velocity autocorrelation function reveals a tendency towards condensation of the cooled supersonic gas, although the nozzles are too small for the formation of clusters. Overall we find that microscopic nozzles act qualitatively like macroscopic nozzles in that the particles are accelerated to supersonic speeds while their thermal motion relative to the stationary flow is cooled. We find that, like macroscopic Laval nozzles, microscopic nozzles also exhibit a sonic horizon, which is well defined on a microscopic scale. The sonic horizon is positioned only slightly further downstream compared to isentropic expansion through macroscopic nozzles, where it is situated in the most narrow part. We analyze the sonic horizon by studying space-time density correlations, i.e., how thermal fluctuations at two positions of the gas density are correlated in time and find that after the sonic horizon there are indeed no upstream correlations on a microscopic scale.

Magnetic field dragging in filamentary molecular clouds

Maps of polarized dust emission of molecular clouds reveal the morphology of the magnetic field associated with star-forming regions. In particular, polarization maps of hub-filament systems show the distortion of magnetic field lines induced by gas flows onto and inside filaments. We aim to understand the relation between the curvature of magnetic field lines associated with filaments in hub-filament systems and the properties of the underlying gas flows. We consider steady-state models of gas with finite electrical resistivity flowing across a transverse magnetic field. We derive the relation between the bending of the field lines and the flow parameters represented by the Alfv\'en Mach number and the magnetic Reynolds number. We find that, on the scale of the filaments, the relevant parameter for a gas of finite electrical resistivity is the magnetic Reynolds number, and we derive the relation between the deflection angle of the field from the initial direction (assumed perpendicular to the filament) and the value of the electrical resistivity, due to either Ohmic dissipation or ambipolar diffusion. Application of this model to specific observations of polarized dust emission in filamentary clouds shows that magnetic Reynolds numbers of a few tens are required to reproduce the data. Despite significant uncertainties in the observations (the flow speed, the geometry and orientation of the filament), and the idealization of the model, the specific cases considered show that ambipolar diffusion can provide the resistivity needed to maintain a steady state flow across magnetic fields of significant strength over realistic time scales.

Unsteady Interaction Between Purge Flow and Secondary Flow in High-Lift Low Pressure Turbine

Abstract The coupling effect between the complex vortex system and the rim purge flow in the endwall region of a high-lift low pressure turbine (LPT) will significantly influence the evolution of secondary flow and the corresponding losses. This paper focused on the unsteady interaction mechanism between the rim purge flow and the secondary flow inside the high-lift LPT under the periodic wake passing. The large eddy simulation (LES) method was used to reveal the influence mechanism of rim purge flow, rotor-stator cavity interaction and unsteady wakes on the secondary flow. Detailed experimental measurement was carried out for the flow field of high-lift LPT under the influences of static purge flow. The results showed that the rim purge flow significantly increased the overturning and underturning downstream of the endwall region, resulting in aggravating secondary loss. The rotating-rim increased the difference of the circumferential velocity between the mainstream and the purge flow, aggravating the Kelvin-Helmholtz (K-H) instability, inducing the K-H vortex structure with stronger energy amplitude, and further increasing the endwall flow loss. The incoming wakes weakened the energy amplitude of the K-H vortex at the rim purge outlet, thinned the thickness of the low-energy fluid at the blade leading edge. Additionally, the interaction between the wakes and the secondary vortices further suppressed the development of the secondary flow. Nevertheless, the incoming wakes caused additional mixing losses and made a negative impact on the overall aerodynamic performance of the high-lift LPT.

Experimental study of simple flumes with converging triangular walls

For water management, the accurate measurement of discharge in irrigation channels is essential. Static flow measuring devices such as weirs and flumes play a significant role in discharge measurement in open channels. Flumes are one of the most commonly used devices for flow measurement. Many researchers have focused on application of simple flumes in irrigation networks. This investigation aims to study the flow discharge through a simple flume with converging triangular walls under free-flow conditions. The flume is constructed by placing two right-angled triangular plates on either side of a rectangular open channel. The channel cross section is rectangular while height of the converging walls reaches to zero at the end of the flume opening. The proposed flume is inexpensive, its operation is simple, its installation is easy and does not require high maintenance (maintenance free). The present study is designed to determine the effect of different variables on flow discharge of this kind of flumes. For this, an experimental program was conducted under upstream subcritical flow regime and under free outflow conditions to formulate the flow discharge of the flume. Flume discharge relationship was deduced using dimensional analysis method and then calibrated using the experimental data collected in this study (318 runs). The proposed discharge equation has an average absolute relative error of 1.66 %, and for 96.5 % of the measured discharge values, the relative errors are within ±5 %. To reliably estimate the flume discharge, free-flow and submerged-flow conditions should be distinguished. For this, suitable equation with an average absolute relative error less than 3.44 % was presented to estimate the submergence threshold. This flume is suitable for the rectangular channel for accurate measurement of free flow at any location on field. This particular device has considerable advantages over the other devices, as it can be quickly installed and removed for use in irrigation channels. The findings and results of this study will be of interest for practical applications.

A Study on the Mechanism of Fracture Initiation and Propagation under Multi-Perforation Conditions in Hydraulic Fracturing

To reveal the mechanism of hydraulic fracture initiation and propagation under the conditions of multiple perforations during horizontal well fracturing, we creatively conducted dual-hole fracturing experiments on small rock samples and established a two-dimensional model of a single cluster with multiple perforations in a horizontal well based on the extended finite element method using the fluid–solid coupling equation, which was combined with the basic theory of damage mechanics. The biggest difference from previous research is that this model does not consider the hypothesis of stress shadows and only focuses on studying the initiation and propagation of multiple perforations in one cluster. We studied the effects of perforation parameters, stress state, and injection flow rate on the initiation and propagation of hydraulic fractures using this model. The experimental and simulation results indicate that under multi-perforation conditions, the number of fractures depends on the number of perforations. The simulation results show that when the spacing between perforations increases or the number of perforations reduces, the initiation time of perforation is advanced and the interference between fractures weakens, which is conducive to the initiation and propagation of hydraulic fractures. As the stress difference increases, the initiation time of perforation becomes earlier and the deflection angle of the outermost fractures becomes smaller, which is conducive to the parallel expansion of the fractures. Moreover, although this has little impact on the morphology of fractures with the rise in flow rate in simulation, it is beneficial for improving the initiation and propagation speed of fractures. The length of fractures also increases significantly at the same time point. In addition, both the experiments and simulations revealed that an increase in the flow rate could accelerate the initiation time of fractures. The proposed model can guide fracturing construction to optimize the design of perforation spacing during horizontal well fracturing, which can contribute to reducing development costs and improving the final production.

An Analytic Element Model for Seepage Forces in Fractured Media

AbstractThere are many situations where groundwater flow has high flow rates, causing large seepage forces. Examples are flows around highly conductive fractures and tunnels. We present a new analytic element for a tunnel in an elastic medium. We combined the analytic element method for groundwater flow with that for linear elasticity, and include the seepage force as a body force in the linearly elastic model. We represent tunnels and fractures as analytic elements. The solution for the case considered is limited to steady state flow and fluid-to-solid coupling. We present examples of the computed seepage forces around a tunnel and a fracture as well as a comparison with another numerical model.

Elliptic and triangular flow of charmonium states in heavy ion collisions

Elliptic and triangular flow of charmonium states in heavy ion collisions

Compressed neural networks for reduced order modeling

Reduced order modeling (ROM) techniques, such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), have been widely used to analyze stationary flows. Neural networks such as autoencoders are effective to reduce the dimension of non-stationary flow, but their larger storage requirements compared to POD and DMD compromise the expectations on ROM. The present work aims at compressing the autoencoder model via two distinctively different approaches, i.e., pruning and singular value decomposition (SVD). The developed algorithm is then applied to reconstruct the flow fields of typical stationary (i.e., a laminar cylinder flow and two turbulent channel flows) and non-stationary (i.e., a laminar co-rotating vortex pair) examples. It is shown that pruning and SVD reduce the size of the autoencoder network to 6% and 3% for the two simple laminar cases (or 18% and 13%, 20%, and 10% for the two complex turbulent channel flow cases), respectively, with approximately the same order of accuracy. Therefore, the proposed autoencoders optimized by the network pruning and SVD lead to effective ROM of both stationary and non-stationary flows although they require more iterations to converge than conventional methods.

Gas state equation and flow mechanism of gas–liquid two-phase flow in airlift pump system

Airlift pumps (ALPs) have a simple structure and significant application potential. However, previous studies on ALPs have generally assumed that the gas density remains constant. In this paper, the gas state equation (GSE) for ALP is established based on the van der Waals formula, which explicitly considers the density of gas. An electrical resistance tomography system is used to collect the gas void fractions at different heights under different gas flow rates, and an empirical formula for the gas void fraction is established. To verify the effectiveness of the proposed model, high-precision pressure sensors and a high-speed camera are used alongside an electrical resistance tomography system to determine the realistic ALP flow parameters. The results of 409 sets of experiments show that: (1) the gas in ALP cannot be regarded as ideal gas, because the ideal GSE cannot distinguish between different gas flow rates; (2) the state change of gas in ALP is a quasi-equilibrium process, whereby the GSE of ALP can be obtained from the pressures under several locations; (3) the axial pressures predicted by the proposed GSE for ALP are in good agreement with experimental data; and (4) a single parameter of the GSE uniquely determines the state process. The proposed model and the experimental data provide a new methodology and comprehensive references for studying the working mechanism and efficiency of ALPs.

A variational data assimilation approach for sparse velocity reference data in coarse RANS simulations through a corrective forcing term

The Reynolds-averaged Navier–Stokes (RANS) equations provide a computationally efficient method for solving fluid flow problems in engineering applications. However, the use of closure models to represent turbulence effects can reduce their accuracy. To address this issue, recent research has explored data-driven techniques such as data assimilation and machine learning.An efficient variational data assimilation (DA) approach is presented to enhance steady-state eddy viscosity based RANS simulations. To account for model deficiencies, a corrective force term is introduced in the momentum equation. In the case of only velocity reference data, this term can be represented by a potential field and is divergence-free. The DA implementation relies on the discrete adjoint method and approximations for efficient gradient evaluation.The implementation is based on a two-dimensional coupled RANS solver in OpenFOAM, which is extended to allow the computation of the adjoint velocity and pressure as well as the adjoint gradient. A gradient-based optimizer is used to minimize the difference between the simulation results and the reference data. To evaluate this approach, it is compared with alternative data assimilation methods for canonical stationary two-dimensional turbulent flow problems. For the data assimilation, sparsely distributed reference data from averaged high-fidelity simulation results are used.The results suggest that the proposed method achieves the optimization goal more efficiently compared to applying data assimilation for obtaining the eddy viscosity, or a field modifying the eddy viscosity, directly. The method works well for different reference data configurations and runs efficiently by leveraging coarse meshes.