Velocity-vorticity Formulation Research Articles

Fluid flow simulation with an [formula omitted]-accelerated Boundary-Domain Integral Method

The development of new numerical methods for fluid flow simulations is challenging but such tools may help to understand flow problems better. Here, the Boundary-Domain Integral Method is applied to simulate laminar fluid flow governed by a dimensionless velocity–vorticity formulation of the Navier–Stokes equation. The Reynolds number is chosen in all examples small enough to ensure laminar flow conditions. The false transient approach is utilised to improve stability.As all boundary element methods, the Boundary-Domain Integral Method has a quadratic complexity. Here, the H2-methodology is applied to obtain an almost linear complexity. This acceleration technique is not only applied to the boundary only part but more important to the domain related part of the formulation. The application of the H2-methodology does not allow to use the rigid body method to determine the singular integrals and the integral free term as done until now. It is shown how to apply the technique of Guigiani and Gigante to handle the strongly singular integrals in this application. Further, a parametric study shows the influence of the introduced approximation parameters. For this purpose the example of a lid driven cavity is utilised. The second example demonstrates the performance of the proposed method by simulating the Hagen–Poiseuille flow in a pipe. The third example considers the flow around a rigid cylinder to show the behaviour of the method for an unstructured grid. All examples show that the proposed method results in an almost linear complexity as the mathematical analysis promisses.

Stability and receptivity analyses of the heated flat-plate boundary layer with variable viscosity

This article presents the modal, non-modal, and resolvent analyses of the boundary layer developed over a heated flat-plate with temperature-dependent viscosity using linear stability theory. The governing equations are derived in the normal velocity–vorticity form by imposing small infinitesimal disturbances on the base flow with the Oberbeck-Boussinesq (OB) approximation. The spectral method is employed to discretize the governing stability equations in the wall-normal direction. The base flow comes from the similarity solution using R-K4th order method along with shooting techniques. The effects of viscosity stratification, inertia, shearing, and buoyancy on the stability of the boundary layer are investigated by varying the sensitivity parameter (ϵ), Reynolds (Re), Prandtl (Pr), and Richardson numbers (Ri). The modal analysis shows the time-asymptotic behavior of the disturbances and onset of instability is mainly caused by amplification of Tollmien–Schlichting (T-S) waves similar to as observe in the traditional Blasius case. The modal stability increases with increase in sensitivity parameter and stabilizing effects become more pronounced for liquid than gas. However, the thermal effects lead to destabilize the flow and strong destabilizing effects produced for higher Prandtl number. On the other hand, non-modal analysis displays an early transient growth of disturbances and an existence of these non-normality effects identified from the pseudospectra via. resolvent analysis. The non-modal growth increases with increase in ϵ even though T-S modes shift towards the damped region, which indicates the continuous modes in the eigenspectrum are more dominated than the discrete T-S modes due to the thermal effects. To clarify this qualitative change, a component-wise input–output analysis is performed to measure the receptivity to particular external disturbances. The results show the thermal energy of the disturbances is converted into kinetic energy due to thermal effects, resulting in strong receptivity amplification at the continuous mode due to the non-normality of the linear operator. Thus, the boundary layer under the influence of viscosity stratification, heating, and shearing effects is vulnerable to free-stream disturbances that could significantly affect bypass transition

Three-dimensional vorticity–velocity formulation in a lattice Boltzmann method

In recent decades, a paradigm shift in macroscopic methods has favored the use of non-primitive variables, such as velocity and vorticity (V–V), over traditional primitive variables. This shift eliminates the need for solving a Poisson equation for pressure, aligning numerical treatments more closely with physical reality. However, the lattice Boltzmann method (LBM), renowned for its efficacy in studying fluid flow phenomena, continues to rely on the conventional pressure–velocity (P–V) approach. This conventional approach necessitates a pressure–density relation, posing challenges in maintaining the incompressible condition. This study pioneers a novel application of the LBM to three-dimensional velocity–vorticity equations, expanding upon our suggested recent method for two-dimensional cases [Kefayati, Phys. Fluids. 36, 013128 (2024)]. To address the complexities introduced by the vortex stretching term in three dimensions, a new equilibrium distribution function is formulated and introduced to the three-dimensional nature of the vorticity vector. The paper details the derivation of the three-dimensional LBM and substantiates its effectiveness through numerical examples, showcasing its applicability in fluid dynamics. By bridging the gap between traditional P–V formulations and the benefits of non-primitive V–V variables, this work contributes to the ongoing exploration of LBM applications in fluid dynamics. The focus on three-dimensional scenarios involving velocity–vorticity equations marks a significant advancement, offering insights into the nuanced dynamics of fluid flow and paving the way for more accurate and realistic simulations in complex environments.

Integration of vorticity–velocity formulation in a lattice Boltzmann method for porous media

A lattice Boltzmann method (LBM) is proposed to address the two-dimensional macroscopic equations of velocity–vorticity for porous media in incompressible flows. The macroscopic equations of two-dimensional porous media using the representative element volume scale approach, employing various models, are shown. The momentum equations are transformed and presented in the velocity and vorticity format. Additionally, the energy and concentration equations are thoroughly examined. Subsequently, the LBM is presented to restore the dimensional macroscopic equations of the velocity–vorticity format for various porous models, accounting for external forces. The paper provides proof and derivations of the equations for the LBM, which are then demonstrated and discussed. The suggested approach is evaluated across a variety of well-established benchmark examples within the realm of fluid flow, heat, and mass transfer in porous media. Importantly, the LBM approach significantly reduces computational time compared to previous methods in the field by eliminating pressure in the momentum equation, thereby reducing the number of unknown variables and transforming the equation into a convection–diffusion form. This modification leads to a linear equilibrium distribution function and a noteworthy decrease in computational costs.

Implementing vorticity–velocity formulation in a finite difference lattice Boltzmann method for two-dimensional incompressible generalized Newtonian fluids

A finite difference lattice Boltzmann approach is introduced to address the two-dimensional macroscopic equations of velocity–vorticity for generalized Newtonian fluids (GNFs). The study involves equations governing macroscopic momentum, energy, and concentration, along with constitutive models applicable to Newtonian, power-law, and viscoplastic fluids. Subsequently, the lattice Boltzmann method, which recovers these macroscopic equations, is detailed, along with proof of its capability to reproduce the aforementioned equations. In order to evaluate the effectiveness and time efficiency of the method, it is validated against various benchmarks. The results demonstrate the efficacy of the proposed method in successfully solving isothermal, thermal, and solutal problems of GNFs, while significantly reducing computational time compared to our previously suggested approach in this domain.

A variational quantum algorithm-based numerical method for solving potential and Stokes flows

This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems. In this method, the governing equations for potential and Stokes flows can be respectively written in the form of Laplace's equation and Stokes equations using velocity potential, stream function and vorticity formulations. Then the finite difference method and the generalized differential quadrature (GDQ) method are applied to discretize the governing equations. For the prescribed boundary conditions, the corresponding linear systems of equations can be obtained. These linear systems are solved by using the variational quantum linear solver (VQLS), which resolves the potential and Stokes flow problems equivalently. To the best of authors' knowledge, this is the first study that incorporates the GDQ method which is inherently a high-order discretization method with the VQLS algorithm. Since the GDQ method can utilize much fewer grid points than the finite difference method to approximate derivatives with a higher order of accuracy, the size of the input matrix for the VQLS algorithm can be smaller. In this way, the computational cost may be saved. The performance of the present method is comprehensively assessed by two representative examples, namely, the potential flow around a circular cylinder and Stokes flow in a lid-driven cavity. Numerical results validate the applicability and accuracy of the present VQLS-based method. Furthermore, its time complexity is evaluated by the heuristic scaling, which demonstrates that the present method scales efficiently in the number of qubits n and the precision ε. This work brings quantum computing to the field of computational fluid dynamics. By virtue of quantum advantage over classical methods, promising advances in solving large-scale fluid mechanics problems of engineering interest may be prompted.

A sharp immersed method for 2D flow-body interactions using the vorticity-velocity Navier-Stokes equations

Immersed methods discretize boundary conditions for complex geometries on background Cartesian grids. This makes such methods especially suitable for two-way coupled flow-body problems, where the body mechanics are partially driven by hydrodynamic forces. However, for the vorticity-velocity form of the Navier-Stokes equations, existing immersed geometry discretizations for two-way coupled problems only achieve first order spatial accuracy near solid boundaries. Here we introduce a sharp-interface approach based on the immersed interface method to handle the one- and two-way coupling between an incompressible flow and one or more rigid bodies using the 2D vorticity-velocity Navier-Stokes equations. Our main contributions are three-fold. First, we develop and analyze a moving boundary treatment for sharp immersed methods that can be applied to PDEs with implicitly defined boundary conditions, such as those commonly imposed on the vorticity field. Second, we develop a two-way coupling methodology for the vorticity-velocity Navier-Stokes equations based on control-volume momentum balance that does not require the pressure field. Third, we show through extensive testing and validation that our resulting flow-body solver reaches second-order accuracy for most practical scenarios, and provides significant efficiency benefits compared to a representative first-order approach.

Numerical simulation of fluid flow with a fast Boundary‐Domain Integral Method

AbstractFor the simulation of fluid flow the velocity‐vorticity formulation of the Navier‐Stokes equations can be used. Approximating the first order time derivative in these equations by finite differences, essentially, the Yukawa equation for the velocity has to be solved with an inhomogeneity governed by the vorticity. Applying the Boundary Element Method to such problems is not straightforward. However, the Boundary‐Domain Integral Method (BDIM) can be used, which requires to solve beside the boundary integrals as well a domain integral due to the inhomogeneity. Classical BEM formulations have a quadratic complexity with respect to the unknowns on the boundary and, for the BDIM additionally, for the unknowns in the domain. This prevents the BDIM from being applied to real‐world problems.To reduce the quadratic complexity in BEM, approximation methods like the fast multipole method, ℋ‐matrices in combination with the adaptive cross approximation or the ℋ2 ‐matrix technique can be used. All these methods reduce the complexity to an almost linear order. Here, the ℋ2 ‐matrix technique is applied to reduce storage and computing time for the right hand side, i.e., for the matrix of the domain integrals. Based on interpolatory kernel expansions and an heuristic criterium for too small matrix entries an almost linear complexity can be achieved. The numerical studies focus on a lid‐driven cavity. The complexity reduction is demonstrated.

High-Order-Accurate Incompressible Flow Solver for High-Fidelity Numerical Simulations and Linear Stability Analysis

The development of high-fidelity versatile incompressible Navier–Stokes solver that is applicable for both direct numerical simulations (DNSs) and linear global stability investigations is presented and discussed. The solver is based on a vorticity–velocity formulation of the Navier–Stokes equations for orthogonal curvilinear grids. A rigorous and efficient approach is proposed that guarantees the divergence-free condition for the velocity and vorticity fields. The code incorporates advanced numerical algorithms including compact finite differences in combination with a pseudo-spectral method, together with an efficient and high-order-accurate Poisson solver. Linear stability modules based on the linearized Navier–Stokes equations (LNSEs) tailored for primary and secondary instability analyses are developed and incorporated into the code, such that the solver can switch from DNS to LNSE in a consistent manner by turning off the nonlinear terms and vice versa. The linear framework includes all the nonparallel effects with respect to the steady/unsteady baseflow and the form of disturbances, and because it is formulated as an initial value problem, both convective and absolute/global instability mechanisms can be investigated. An additional significant advantage of the linear module is that, for the secondary stability analysis, time-periodic assumption is not required and/or the unsteady baseflow does not have to be computed a priori. In the present paper, results obtained with the new Navier–Stokes solver are compared with benchmark solutions for the flow past a static and oscillating circular cylinder. The new code then was employed for a three-dimensional DNS of the flow for a wing section with a modified NACA 6 series airfoil at a chord Reynolds number of . Furthermore, the capability of the linear stability modules was demonstrated by investigating the primary and secondary (convective and absolute) instability mechanisms for boundary layers and the flow past a circular cylinder.

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by nonconvex immersed bodies, multiply connected domains, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the immersed solid bodies, reducing computational cost and enabling simulations with nonconvex bodies that could not be discretized with previous immersed interface methods. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, allowing for simulations with multiply connected domains and outflow boundaries that have challenged other immersed interface vortex methods. We also explore novel methods for recovering time-dependent pressure distributions on immersed bodies within a vorticity-based method and present a novel control volume formulation for recovering aerodynamic moments from only the vorticity and velocity fields. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains.

Impact of nanoparticle shape on thermo-solutal buoyancy induced lid-driven-cavity with inclined magnetic-field

The aim of this paper is to numerically investigate the influence of nanoparticles shape on heat and mass transport phenomena in a moving lid cavity under the combined effect of thermo-solutal buoyancy force and magnetic force. The governing equations are transformed into velocity-vorticity form of equations and solved using Galerkin's weighted residual finite-element-technique. The analysis has been carried out with parameters like buoyancy ratio (−5 ≤ N ≤ 5), magnetic field inclination angle (0° ≤ ϕ ≤ 90°) with four shapes of Al2O3 nanoparticle like bricks, blades, platelets and cylinders. The results revealed that the shape factor on Nusselt number is significant depending upon the inclined magnetic field and buoyancy ratio whereas on mass transfer the shape effect is negligible. The diffusion mode of transport process is stronger than the convection mode at higher inclination angle of magnetic field. Based on a given value of N and ϕ, blade and cylinder shows the best performance in Nusselt and Sherwood number respectively except the platelet shape that shows maximum frictional loss in terms of wall shear stress.

Double diffusive mixed convection in a Cu-Al2O3/water nanofluid filled backward facing step channel with inclined magnetic field and part heating load conditions

The impact of inclined magnetic field on thermo-fluidic behaviour in a Cu-Al2O3/water nanofluid filled backward facing step channel, associated with thermo-solutal buoyancy induced mixed convection, has been numerically investigated. A novel part heating load boundary condition is applied at the bottom wall in order to model and analyse closer to the actual physics. The Navier-stokes equation in velocity-vorticity form along with energy and concentration equations are solved using Galerkin's FEM. The variation of buoyancy ratio (‒5≤N≤5), Hartmann number (0≤Ha≤50), inclination angle (0≤φ≤90) with two different part heating arrangements are analyzed systematically to understand the flow pattern and convective heat & mass transfer performance. Results show that at positive buoyancy ratio the solutal buoyancy force always supports the thermal buoyancy force augmenting the heat and mass transport phenomena. Increasing the inclination angle always give rise to the conductive mode of heat transfer compared to convective mode caused by the presence of stronger magnetic force. At N=5, change in Ha from 0 to 50 shows a maximum increment of 38% in Nuavg for φ= 0° wheras 5% reduction in Shavg is obtained for φ= 30°. Compared to case II, nearly 20 to 40% improvement in convective heat transfer is observed in case-I where heat sources are located closer to the channel step though hardly any change is observed in convective mass transfer between two different cases.

Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives

In fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-dimensional fourth-order elliptic problems. An efficient method to solve these two-dimensional bi-Laplacian operators with Neumann homogeneus boundary conditions was designed and validated using 2D compact finite difference schemes. The solution is formulated as a linear combination of auxiliary solutions, as many as the number of points on the boundary, a method that was prohibitive some years ago due to the large memory requirements to store all these auxiliary functions. The validation has been made for different field configurations, grid sizes, and stencils of the numerical scheme, showing its potential to tackle high gradient fields as those that can be found in turbulent flows.

An adjoint approach for computing the receptivity of the rotating disc boundary layer to surface roughness

An adjoint approach is developed to compute the receptivity of the rotating disc boundary layer to surface roughness. The adjoint linearised Navier–Stokes equations, in cylindrical coordinates, are derived and receptivity characteristics are computed for a broad range of azimuthal mode numbers using a fully equivalent velocity–vorticity formulation. For each set of flow conditions (i.e. azimuthal mode number), the adjoint method only requires that the linear and adjoint solutions be computed once. Thus, the adjoint approach offers significant computational and time advantages over alternative receptivity schemes (i.e. direct linearised Navier–Stokes) as they can be used to instantaneously compute the receptivity of boundary layer disturbances to many environmental mechanisms. Stationary cross-flow disturbances are established by randomly distributed surface roughness that is periodic in the azimuthal direction and modelled via a linearisation of the no-slip condition on the disc surface. Each roughness distribution is scaled on its respective root-mean-square. A Monte-Carlo type uncertainty quantification analysis is performed, whereby mean receptivity amplitudes are computed by averaging over many thousands of roughness realisations with variable length and wavelength filters. The amplitude of the cross-flow instability is significantly larger for roughness distributions near the conditions for neutral linear instability, while roughness elements radially outboard have a negligible effect on the receptivity process. Furthermore, receptivity increases sharply for roughness distributions that encompass wavelength scales equivalent to that associated with the cross-flow instability. Finally, mean receptivity characteristics are used to predict the radial range that stationary cross-flow vortices achieve amplitudes sufficient to invalidate the linear stability assumptions.

Control of stationary convective instabilities in the rotating disk boundary layer via time-periodic modulation

The control of stationary convective instabilities in the rotating disk boundary layer via a time-periodic modulation of the disk rotation rate is investigated. The configuration provides an archetypal example of a three-dimensional temporally periodic boundary layer, encompassing both the von Kármán and Stokes boundary layers. A velocity–vorticity formulation of the governing perturbation equations is deployed, together with a numerical procedure that utilises the Chebyshev-tau method. Floquet theory is used to determine the linear stability properties of these time-periodic flows. The addition of a time-periodic modulation to the otherwise steady disk rotation rate establishes a stabilising effect. In particular, for a broad range of modulation frequencies, the growth of the stationary convective instabilities is suppressed and the critical Reynolds number for the onset of both the cross-flow and Coriolis instabilities is raised to larger values than that found for the steady disk without modulation. An energy analysis is undertaken, where it is demonstrated that time-periodic modulation induces a reduction in the Reynolds stress energy production and an increase in the viscous dissipation across the boundary layer. Comparisons are made with other control techniques, including distributed surface roughness and compliant walls.

Numerical investigation of double-diffusive mixed convection of Fe3O4/Cu/Al2O3-water nanofluid flow through a backward-facing-step channel subjected to magnetic field

PurposeThis study aims to investigate the buoyancy-induced heat and mass transfer phenomena in a backward-facing-step (BFS) channel subjected to applied magnetic field using different types of nanofluid.Design/methodology/approachConservation equations of mass, momentum, energy and concentration are used through velocity-vorticity form of Navier–Stokes equations and solved using Galerkin’s weighted residual finite element method. The density variation is handled by Boussinesq approximation caused by thermo-solutal buoyancy forces evolved at the channel bottom wall having high heat and concentration. Simulations were carried out for the variation of Hartmann number (0 to 100), buoyancy ratio (−10 to +10), three types of water-based nanofluid i.e. Fe3O4, Cu, Al2O3 at χ = 6%, Re = 200 and Ri = 0.1.FindingsThe mutual interaction of magnetic force, inertial force and nature of thermal-solutal buoyancy forces play a significant role in the heat and mass transport phenomena. Results show that the size of the recirculation zone increases at N = 1 for aiding thermo-solutal buoyancy force, whereas the applied magnetic field dampened the fluid-convection process. With an increase in buoyancy ratio, Al2O3 nanoparticle shows a maximum 54% and 67% increase in convective heat and mass transfer, respectively at Ha = 20 followed by Fe3O4 and Cu. However, with increase in Ha the Nuavg and Shavg diminish by maximum 62.33% and 74.56%, respectively, for Fe3O4 nanoparticles at N = 5 followed by Al2O3 and Cu.Originality/valueThis research study numerically examines the sensitivity of Fe3O4, Cu and Al2O3 nanoparticles in a magnetic field for buoyancy-induced mixed convective heat and mass transfer phenomena in a BFS channel, which was not analyzed earlier.

Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic equations for vorticity. Probabilistic representations for solutions of these linear equations are given. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot--Savart-type law. We show that the approximation is divergent free and of first order. The results are extended to two-dimensional stochastic Navier-Stokes equations with additive noise, where, in particular, we prove the first mean-square convergence order of the vorticity approximation.

Investigation of grid-based vorticity-velocity large eddy simulation off-body solvers for application to overset CFD

Accurately predicting unsteady wakes and vortex-dominated flows is essential to a wide range of engineering applications, including aircraft, rotorcraft, shipboard operations, bio-inspired unsteady flight and propulsion, wind turbines, and urban flows. While current CFD software can model the complete flow field and wake system, the computational costs incurred in high Reynolds number unsteady turbulent flow simulations often remain prohibitive for routine engineering use, particularly for applications involving moving components. Prior work has demonstrated that by adopting a vorticity-velocity formulation in a grid-based off-body flow solver (VorTran-M and VorTran-M2) one can lower these costs by several orders of magnitude when compared to conventional approaches. This paper describes the extensions made to VorTran-M2 to support turbulent flows, and associated benchmarking activity to assess its performance for problems involving strong stretching and diffusion processes, whose competing contributions to the vorticity field are core drivers of turbulent flow evolution. Predictions are presented for: (i) the Kida-Pelz problem whose inviscid form is of mathematical interest due to its apparent formation of singular flow in finite time; and (ii) the Taylor Green vortex arrangement, which has been extensively studied as a fundamental simulation challenge in the turbulence modeling community. The results are used to evaluate the overall predictive ability and performance of two sub-grid scale models incorporated into VorTran-M2. Results indicate that the computational cost savings seen previously for inviscid and convection dominated problems extend to turbulent flow simulations supporting the viability of VorTran-M2 as a low cost means for accurately modeling far-field and background flows, particularly when long duration vorticity evolution is of interest.

A Review of Vortex Methods and Their Applications: From Creation to Recent Advances

This review paper presents an overview of Vortex Methods for flow simulation and their different sub-approaches, from their creation to the present. Particle methods distinguish themselves by their intuitive and natural description of the fluid flow as well as their low numerical dissipation and their stability. Vortex methods belong to Lagrangian approaches and allow us to solve the incompressible Navier-Stokes equations in their velocity-vorticity formulation. In the last three decades, the wide range of research works performed on these methods allowed us to highlight their robustness and accuracy while providing efficient computational algorithms and a solid mathematical framework. On the other hand, many efforts have been devoted to overcoming their main intrinsic difficulties, mostly relying on the treatment of the boundary conditions and the distortion of particle distribution. The present review aims to describe the Vortex methods by following their chronological evolution and provides for each step of their development the mathematical framework, the strengths and limits as well as references to applications and numerical simulations. The paper ends with a presentation of some challenging and very recent works based on Vortex methods and successfully applied to problems such as hydrodynamics, turbulent wake dynamics, sediment or porous flows.

NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations

In the last 50 years there has been a tremendous progress in solving numerically the Navier-Stokes equations using finite differences, finite elements, spectral, and even meshless methods. Yet, in many real cases, we still cannot incorporate seamlessly (multi-fidelity) data into existing algorithms, and for industrial-complexity applications the mesh generation is time consuming and still an art. Moreover, solving ill-posed problems (e.g., lacking boundary conditions) or inverse problems is often prohibitively expensive and requires different formulations and new computer codes. Here, we employ physics-informed neural networks (PINNs), encoding the governing equations directly into the deep neural network via automatic differentiation, to overcome some of the aforementioned limitations for simulating incompressible laminar and turbulent flows. We develop the Navier-Stokes flow nets (NSFnets) by considering two different mathematical formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. Since this is a new approach, we first select some standard benchmark problems to assess the accuracy, convergence rate, computational cost and flexibility of NSFnets; analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. This is unsupervised learning and, hence, no labeled data are required beyond boundary and initial conditions and the fluid properties. The residuals of the VP or VV governing equations, together with the initial and boundary conditions, are embedded into the loss function of the NSFnets. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without extra computational cost. Unlike the traditional numerical methods, NSFnets inherit the properties of neural networks (NNs), hence the total error is composed of the approximation, the optimization, and the generalization errors. Here, we empirically attempt to quantify these errors by varying the sampling (“residual”) points, the iterative solvers, and the size of the NN architecture. For the laminar flow solutions, we show that both the VP and the VV formulations are comparable in accuracy but their best performance corresponds to different NN architectures. The initial convergence rate is fast but the error eventually saturates to a plateau due to the dominance of the optimization error. For the turbulent channel flow, we show that NSFnets can sustain turbulence at Reτ∼1,000, but due to expensive training we only consider part of the channel domain and enforce velocity boundary conditions on the subdomain boundaries provided by the DNS data base. We also perform a systematic study on the weights used in the loss function for balancing the data and physics components, and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. In the last part, we demonstrate how NSFnets should be used in practice, namely for ill-posed problems with incomplete or noisy boundary conditions as well as for inverse problems. We obtain reasonably accurate solutions for such cases as well without the need to change the NSFnets and at the same computational cost as in the forward well-posed problems. We also present a simple example of transfer learning that will aid in accelerating the training of NSFnets for different parameter settings.