Three-dimensional Incompressible Navier-Stokes Equations Research Articles

The Minkowski dimension of boundary singular points in the Navier–Stokes equations

We study the partial regularity problem of the three-dimensional incompressible Navier–Stokes equations. We present a new boundary regularity criterion for boundary suitable weak solutions. As an application, a bound for the parabolic Minkowski dimension of possible singular points on the boundary is obtained.

Ceiling effects on the aerodynamics of a flapping wing at hovering condition

The ceiling effect on the aerodynamics of a hovering flapping wing is investigated by solving the three-dimensional incompressible Navier-Stokes equations. Computations have been carried out for some parameters including the distance between the wing and the ceiling, the Reynolds number, the stroke amplitude, and the mid-stroke angle of incidence. The ceiling effect on the force production and vortical structures around the wing is analyzed. It is shown that the ceiling effect increases the aerodynamic forces. This improvement in force production in the ceiling effect is caused by the increments both in the relative velocity of oncoming flow and the effective angle of attack of the wing. The underlying mechanism is that the presence of the ceiling acts as a mirror as if there exists a mirroring leading-edge vortex (LEV). This mirroring LEV not only increases the relative velocity of the oncoming flow ahead of the wing but also produces an upwash to the oncoming flow, hence increasing the effective angle of attack of the wing.

An improved mixed Lagrangian–Eulerian (IMLE) method for modelling incompressible Navier–Stokes flows with CUDA programming on multi-GPUs

In this study, a GPU-accelerated improved mixed Lagrangian–Eulerian (IMLE) method is proposed to solve the three-dimensional incompressible Navier–Stokes equations. To improve the prediction accuracy, the proposed IMLE method approximates the total derivative term in Lagragian sense, and the spatial derivative terms are approximated on Eulerian coordinates. Transfer of data from Lagrangian particles to data on Eulerian grids is accurately carried out by adopting moving least squares (MLS) interpolation method. The velocity-pressure decoupling issue is overcome by adopting pressure-free projection method in which the pressure field is calculated by solving a pressure Poisson equation (PPE). It is noted that the MLS interpolation is time consuming since this procedure belongs to a pointwise scheme in which a local matrix equation shall be solved on each grid point. In addition, the discretized PPE forms a large sparse matrix and it is computationally intensive to solve by using the conjugate gradient (CG) method. Therefore, we are aimed to resort to CUDA- and OpenMP-programming means to accelerate the computation. In this study, the performance of the multiple GPUs code can reach up to 27 times faster with respect to multi-threads CPU performance.

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, ut+uxxxx+∂xxux2=0, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier–Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder Q if the Serrin condition ux∈Lq′Lq(Q) is satisfied, where q,q′∈[1,∞] are such that either 1/q+4/q′<1 or 1/q+4/q′=1, q′<∞.

Properties of Stationary Statistical Solutions of the Three-Dimensional Navier–Stokes Equations

The stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier–Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray–Hopf weak solutions which are locally strong solutions. Two recurrence-type results related to these measures are also proved.

Numerical simulation of multiple steady and unsteady flow modes in a medium-gap spherical Couette flow

We study the multiple steady and unsteady flow modes in a medium-gap spherical Couette flow (SCF) by solving the three-dimensional incompressible Navier–Stokes equations. We have used an artificial compressibility method with an implicit line Gauss–Seidel scheme. The simulations are performed in SCF with only the inner sphere rotating. A medium-gap clearance ratio, $$\sigma =\left( R_{2}-R_{1}\right) /R_{1}=0.25,$$ has been used to investigate various flow states in a range of Reynolds numbers, $${Re}\in [400,6500]$$ . First, we compute the 0-vortex basic flow directly from the Stokes flow as an initial condition. This flow exists up to $${Re}=4900$$ after which it evolves into spiral 0-vortex flows with wavenumber $$s_p=3,4$$ in the range $${Re} \in [4900,6000]$$ , and then the flows become turbulent when $${Re}>6000$$ . Second, we obtain the steady 1-vortex flow by using the 1-vortex flow at $${Re} =700$$ for $$\sigma =0.18$$ as the initial conditions and found that it exists for $${Re} \in [480,4300]$$ . The 1-vortex flow becomes wavy 1-vortex in the range $${Re} \in [4400,5000]$$ . Further increasing the Reynolds number, we obtain new spiral waves of wavenumber $$s_p=3$$ for $${Re}\in [5000, 6000]$$ . The flow becomes turbulent when $${Re}>6000$$ . Third, we obtain the steady 2-vortex flow by using the 2-vortex flow at $${Re} =900$$ for $$\sigma =0.18$$ as the initial conditions and found that it exists for $${Re} \in [700,1900]$$ . With increasing Reynolds number the 2-vortex flow becomes partially wavy 2-vortex in the small range $${Re} \in [1900,2100]$$ . We obtain distorted spiral wavy 2-vortex in the range $${Re} \in [4000,5000]$$ . when $${Re}>6000$$ the flow evolves into spiral 0-vortex flow and becomes turbulent. The present flow scenarios with increasing Re agree well with the experimental results and further we obtain new flow states for the 1-vortex and 2-vortex flows.

Direct Numerical Simulation of Gas–Particle Flows with Particle–Wall Collisions Using the Immersed Boundary Method

We investigated particulate flows by coupling simulations of the three-dimensional incompressible Navier–Stokes equation with the immersed boundary method (IBM). The results obtained from the two-way coupled simulation were compared with those of the one-way simulation, which is generally applied for clarifying the particle kinematics in industry. In the present flow simulation, the IBM was solved using a ghost–cell approach and the particles and walls were defined by a level set function. Using proposed algorithms, particle–particle and particle–wall collisions were implemented simply; the subsequent coupling simulations were conducted stably. Additionally, the wake structures of the moving, colliding and rebounding particles were comprehensively compared with previous numerical and experimental results. In simulations of 50, 100, 200 and 500 particles, particle–wall collisions were more frequent in the one–way scheme than in the two-way scheme. This difference was linked to differences in losses in energy and momentum.

Parametric study of unsteady-flow-induced volute casing vibro-acoustics in a centrifugal fan

A numerical parametric analysis of a vibro-acoustic coupling method that considered the influence of vibro-acoustic coupling was carried out to investigate the casing vibrations and feathers of vibrational noise induced by unsteady flow of the centrifugal fan at the best-efficiency point (BEP). There are three important aspects of this method. First, an unsteady flow-field with a whole impeller-volute configuration was solved based on three-dimensional incompressible Navier-Stokes equations and a standard k-ε turbulence mode to obtain the source of the vibro-acoustics. Second, a one-way-flow structural acoustic coupling method was implemented to study the volute vibrations and behaviors of vibrational noise by adoption. The generation mechanism of vibrational noise of the volute casing was revealed. Third, the parametric analysis method was used to explore the parametric relationship between the panel thicknesses (such as front-panel thickness [FT], side-panel thickness [ST], and back-panel thickness [BT]) and the outlet acoustical power of the volute casing surface. The parametric analysis provides a reasonable range of values of three panel thicknesses that result in minimal vibrational sound radiation.

Investigating the performance of a twisted modified Savonius rotor

This investigation aims to study experimentally and numerically the performance of a twisted modified Savonius rotor at various twisting angles. The experiments are conducted in front of a free air jet. The computations are performed using the three-dimensional incompressible unsteady Reynolds-Average Navier-Stokes (RANS) equations along with the RNG k-ε turbulence model. The comparison with the present measurements shows good prediction of the RNG k-ε turbulence model to the rotor performance. Furthermore, the power coefficient (CP) and the torque coefficient (CT) at different tip speed ratios (λ) for twist angles ranging from 0° to 180° are predicted under constant aspect ratio of 1.0 and overlap ratio of 0.15. The numerical results show that the maximum power coefficient (CPmax) increases with the increase of twist angle up to an optimum value of 45° and then decreases except for twist angle of 135°. The twisted modified rotor with a twist angle of 45° has the highest CP and CT(0.22 and 0.41, respectively) compared to the other studied twist angles.

Effects of Discrete Energy and Helicity Conservation in Numerical Simulations of Helical Turbulence

Helicity is the scalar product between velocity and vorticity and, just like energy, its integral is an inviscid invariant of the three-dimensional incompressible Navier-Stokes equations. However, space- and time-discretization methods typically corrupt this property, leading to violation of the inviscid conservation principles. This work investigates the discrete helicity conservation properties of spectral and finite-differencing methods, in relation to the form employed for the convective term. Effects due to Runge-Kutta time-advancement schemes are also taken into consideration in the analysis. The theoretical results are proved against inviscid numerical simulations, while a scale-dependent analysis of energy, helicity and their non-linear transfers is performed to further characterize the discretization errors of the different forms in forced helical turbulence simulations.

Development of a finite element flow solver for solving three-dimensional incompressible Navier–Stokes solutions on multiple GPU cards

In this paper a multi-GPU-based finite element flow solver is developed to solve the three-dimensional incompressible Navier–Stokes equations at steady-state. To circumvent the convective instability problem at high Reynolds numbers, the proposed streamline upwinding finite element model minimizes the wavenumber error for the convection terms. Mixed finite element formulation is adopted and the resulting nearly ill-conditioned finite element equations are solved iteratively. To avoid the Lanczos or pivoting breakdown, the finite element equations are first normalized. The computationally efficient preconditioned conjugate gradient (PCG) solver can then be applied to get the unconditionally convergent solution. The developed finite element code implemented on multi-GPU cards will be verified and validated by solving the problem amenable to analytical solution and the benchmark lid-driven cavity problem, respectively.

Existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow

We study the existence regime of symmetric and asymmetric Taylor vortices in wide-gap spherical Couette flow by time marching the three-dimensional incompressible Navier–Stokes equations numerically. Three wide-gap clearance ratios, $$\beta =\left( R_{2}-R_{1}\right) /R_{1}=0.33$$ , 0.38 and 0.42 are investigated for a range of Reynolds numbers respectively. Using the 1-vortex flow for clearance ratio $$\beta =0.18$$ at Reynolds number $${Re}=700$$ as the initial conditions and suddenly increasing $$\beta$$ to the target value, we can compute Taylor vortices for the three wide gaps. For $$\beta =0.33$$ , Taylor vortices exist in the range $$450\le {Re}\le 2050$$ . With increasing Re the steady symmetric 1-vortex flow becomes steady asymmetric at $${Re}=1850$$ , and then become periodic at $${Re}=2000$$ . When $${Re}>2050$$ the flow returns back to the steady basic flow state with no Taylor vortices. For $$\beta =0.38$$ , Taylor vortices can exist in the range $$500\le {Re}\le 1400$$ . With increasing Re, the steady symmetric 1-vortex flow become steady asymmetric at $${Re}=1200$$ , and then the flow evolves into the steady basic flow for $${Re}>1400$$ . For $$\beta =0.42$$ , Taylor vortices can exist in the range $$650\le {Re}\le 1300$$ . With increasing Re, steady asymmetric Taylor vortices occur at $${Re}=1150$$ , and then the flow evolves into the steady basic flow for $${Re}>1300$$ . The present numerical results are in good agreement with available numerical and experimental results. Furthermore, the existence regime of Taylor vortices in the $$(\beta ,{Re})$$ plane for $$\beta \ge 0.33$$ and the three-dimensional transition process from periodic asymmetric vortex flow to steady basic flow with increasing Re are presented for the first time.

Maximum palinstrophy amplification in the two-dimensional Navier–Stokes equations

We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier–Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number $Re$ and palinstrophy ${\mathcal{P}}$ which maximize $\text{d}{\mathcal{P}}/\text{d}t$ is constructed by numerically solving suitable optimization problems for a wide range of $Re$ and ${\mathcal{P}}$, providing numerical evidence for the sharpness of the analytic estimate $\text{d}{\mathcal{P}}/\text{d}t\leqslant (a+b\sqrt{\ln Re+c}){\mathcal{P}}^{3/2}$ with respect to both $Re$ and ${\mathcal{P}}$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations, and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e. $\sup _{t&gt;0}{\mathcal{P}}(t)$, are observed to scale with the magnitude of the initial palinstrophy ${\mathcal{P}}(0)$ in accord with the corresponding a priori estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equation are discussed.

A numerical analysis of casing groove parameters on the performance of wave energy conversion device

Casing grooves in Wells turbine can enhance its performance. The present computational work has been carried out to understand the effect of casing-groove's width-to-chord (w/C) ratio on the performance of a wave energy conversion device. In this study, four different values of w/C ratio such as 0.24, 0.48, 0.64 and 0.96 were investigated. The circumferential casing groove depth (3% of the chord length) and tip clearance (1% of the chord length) were kept constant for all the cases. The flow field was analysed through discretisation of three-dimensional steady incompressible Reynolds-averaged Navier-stokes (RANS) equations with two-equation eddy viscosity turbulence closer model. The simulation results show that the w/C ratio has a significant impact on the turbine performance and the w/C = 0.64 was found to be the optimal one among the several values investigated in the present study.

Study on Aerodynamic Shape Optimization of Metro Running in the Open Air

Based on the three-dimensional incompressible Navier-Stokes equations and the standard k~ε two-equation turbulence model, the flow field around the train is simulated numerically in the case of no wind condition. The influence of the head type and the windshield structure on the flow field distribution and the aerodynamic drag coefficient and the lift coefficient are compared and analyzed. The results show that the aerodynamic performance of the streamlined train is better, the resistance of the streamlined train is obviously reduced, especially the first car, tail car resistance decreased significantly, vehicle resistance decreased by 45.9%. The fully enclosed windshield eliminates the flow separation at the windshield and helps to reduce the resistance of the intermediate car, reducing the percentage by 28.6%.

Asymptotic expansion in Gevrey spaces for solutions of Navier–Stokes equations

In this paper, we study the asymptotic behavior of solutions to the threedimensional incompressible Navier-Stokes equations (NSE) with periodic boundary conditions and potential body forces. In particular, we prove that the Foias-Saut asymptotic expansion for the regular solutions of the NSE in fact holds in all Gevrey classes . This strengthens the previous result obtained in Sobolev spaces by Foias-Saut. By using the Gevrey-norm technique of Foias-Temam, the proof of our improved result simplifies the original argument of Foias-Saut, thereby, increasing its adaptability to other dissipative systems. Moreover, the expansion is extended to Leray-Hopf weak solutions and the exponential decay rate for each of its remainders is calculated explicitly.

Some problems on the regularity of solutions for the 3D Navier-Stokes equations

In this paper, we review some recent work on the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations in recent years. It is well known that the three-dimensional incompressible Navier-Stokes system has the global Leray-Hopf weak solutions. When the weak solution satisfies the Prodi-Serrin condition, the solution is regular. We obtain some new results in the regularity conditions. Especially, for axisymmetric system, when the rotation speed is zero, it is well known that the system is globally well-posed. We find a new conserved quantity, and obtain some new advances in the regularity conditions of axisymmetric solutions with nonzero swirl, also get the global well-posedness results with the small initial rotating speed. Furthermore, we also get the similar result for the inhomogeneous system. In the end, we consider the global well-posedness of a class of generalized Navier-Stokes systems with the higher order horizontal viscosity $D_h^{2\alpha}$, $\alpha\geq\frac43$.

Global well-posedness of incompressible Navier-Stokes equations with two slow variables

In this paper, the global well-posedness of the three-dimensional incompressible Navier-Stokes equations with a linear damping for a class of large initial data slowly varying in two directions are proved by means of a simpler approach.

Performance Analysis of a Helical Savonius Wind Turbine with Modified Rotor

This study aims to investigate numerically the performance of a helical Savonius wind rotor with a twist angle of 45°. The twist is performed on a modified profile of the rotor blade. The power coefficient (Cp), the torque coefficient (CT), and the static torque coefficient (CTS) are computed under a constant aspect ratio of 1.0 and overlap ratio of 0.15. The numerical results presented in this paper are obtained using the three-dimensional incompressible unsteady Reynolds-Averaged Navier-Stokes equations along with the RNG k-e turbulence model. A comparison between the present helical Savonius wind turbine of modified rotor and the helical Savonius wind turbine of conventional rotor shows better performance for the present modification. The maximum power coefficient and the maximum torque coefficient for the present helical modified rotor are 0.22 and 0.41, respectively.

Existence and Stability of Spatial Plane Waves for the Incompressible Navier–Stokes in $$\mathbb {R}^3$$ R 3

We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a $L^\infty$ family of global spatial plane wave solutions, which are connected with the two-dimensional equation. We then proceed to prove local well-posedness over a space which includes $L^3(\mathbb{R}^3)$ and these solutions. Finally, we prove $L^3$-stability of spatial plane waves, with no condition on their size.