Axisymmetric Navier-Stokes Equations Research Articles

Fine large-time asymptotics for the axisymmetric Navier–Stokes equations

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-based entropy methods.

Axisymmetric flow model of cannabis oil extraction of supercritical fluid extraction CO2 process

Supercritical fluid extraction using CO2 is a widely recognized method for purifying oil extracts. Understanding the extraction process is important for improving yields. While most research has focused on one-dimensional simulations, there is a need for three-dimensional models that better simulate real extraction behavior. This paper focuses on a two-dimensional axisymmetric model to study the complexities of three-dimensional extraction processes. The proposed mathematical and axisymmetric model describes the extraction of cannabis using a supercritical fluid extraction (SFE) CO2 process. The COMSOL program was used to solve 2D axisymmetric and laminar flow dynamics within the extractor. The velocity was derived from the axisymmetric Navier-Stokes equation was used in the mathematical model. A 2D mathematical and axisymmetric model was developed based on 1D Reverchon models, and it was validated using experimental data. The experimental data were used to determine one unknown parameter. The total mass transfer coefficient (K) was determined from the experimental results and integrated into computational model. Creating a computational model, meshing, and conducting a 2D axisymmetric simulation analysis of Cannabis sativa L. oil extraction. The proposed curve was computed using a numerical method and was adjusted iteratively to converge with the experimental data, verified by minimizing the Root Mean Square (RMS) error. The experimental data yielded a total mass transfer coefficient (K) of 1.0194 x 10-8 m/s, which was then used in the 2D axisymmetric simulation. The results of the axisymmetric simulation show the concentration distribution within the extractor. The predictions of the proposed numerical 2D solutions closely aligned with the experimental data. The concentration of oil in the axisymmetric results is in accordance with the velocity. The model used in the supercritical fluid extraction process is ideal for purifying extractions, especially for medical and pharmaceutical purposes. This model can be further developed into more complex process and new design extractor for achieving high performance extraction. The scientific community can benefit from the model that combines the Navier-Stoke equation and extraction equation. It can be used to improve extraction simulations for other extraction methods or supercritical fluid in different fields.

Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinder

In this paper, we study the initial boundary value problem and vanishing viscosity limit for incompressible axisymmetric Navier–Stokes equations with swirls in the exterior of a cylinder under Navier-slip boundary condition. In the first part, we prove the existence of a unique global solution with the axisymmetric initial data u0ν∈Lσ2(Ω) and axisymmetric force f∈L2([0,T];L2(Ω)) . This result improves the initial regularity condition on the global well-posedness result obtained by K Abe and G Seregin (2020 Proc. R. Soc. Edinburgh A 150 1671–98) and extends their boundary condition. In the second part, we make the first attempt to investigate the inviscid limit of unforced viscous axisymmetric flows with swirls and prove that the viscous axisymmetric flows with swirls converge to inviscid axisymmetric flows without swirls under the condition ‖ru0θν‖L2(Ω)=O(ν) . Some new uniform estimates, independent of the viscosity, are obtained here. The second result can be thought as a follow-up work to the previous work by K Abe (2020 J. Math. Pures Appl. 137 1–32), where the inviscid limit for the same equations without swirls in an infinite cylinder was studied.

Inertia and slip effects on the instability of a liquid film coated on a fibre

To investigate the influence of inertia and slip on the instability of a liquid film on a fibre, a theoretical framework based on the axisymmetric Navier–Stokes equations is proposed via linear instability analysis. The model reveals that slip significantly enhances perturbation growth in viscous film flows, whereas it exerts minimal influence on flows dominated by inertia. Moreover, under no-slip boundary conditions, the dominant instability mode of thin films remains unaltered by inertia, closely aligning with predictions from a no-slip lubrication model. Conversely, when slip is introduced, the dominant wavenumber experiences a noticeable reduction as inertia decreases. This trend is captured by an introduced lubrication model with giant slip. Direct numerical simulations of the Navier–Stokes equations are then performed to further confirm the theoretical findings at the linear stage. For the nonlinear dynamics, no-slip simulations show complex vortical structures within films, driven by fluid inertia near surfaces. Additionally, in scenarios with weak inertia, a reduction in the volume of satellite droplets is observed due to slip, following a power-law relationship.

Numerical investigation of effect of the design parameters of the counter-flow jet for drag reduction in hypersonic low-Reynolds number regime

Due to the correlation of design parameters of the counter-flow jet in addition to the complexity of the flow field, understanding the mechanism of the counter-flow jet for drag reduction and flow control remains challenging. Furthermore, to satisfy the demands of the space transportation system, investigating the counter-flow jet's suitability for a range of flight conditions is critical. To solve these problems, a study was performed by varying the pressure ratio (PR) and exit Mach number of the counter-flow jet at hypersonic low-Reynolds number regime. For numerical simulations, laminar, axisymmetric Navier–Stokes equations were solved by the total variation diminishing scheme with second-order accuracy in space and the explicit strong stability preserving the Runge–Kutta method. With given numerical conditions, the flow field was categorized as the long penetration mode (LPM) based on the penetration length and the fluctuation of the flow field at the high-Reynolds number regime. By reducing the free-stream flow Reynolds number while keeping other parameters unchanged, the flow field transitioned from the LPM to a stable LPM, short penetration mode, or long penetration with periodically oscillation mode. The critical Reynolds number for the transition of the flow field is highly dependent on the exit Mach number of the counter-flow jet and PR. The extended jet layer was the primary reason for the fluctuation of the drag coefficient. Furthermore, with the counter-flow jet at certain flight conditions, drag can be reduced by up to 78% regardless of the stability of the flow field.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Regularity criteria of the axisymmetric Navier-Stokes equations and Hardy-Sobolev inequality in mixed Lorentz spaces

In this paper, we refine some known regularity criteria of the 3D axisymmetric Navier-Stokes equations via establishing several new Hardy-Sobolev inequalities in mixed Lebesgue spaces and mixed Lorentz spaces. These inequalities cover many known corresponding results and are of independent interest.

Quantitative Control of Solutions to the Axisymmetric Navier-Stokes Equations in Terms of the Weak L3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^3$$\\end{document} Norm

We are concerned with strong axisymmetric solutions to the 3D incompressible Navier-Stokes equations. We show that if the weak L3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^3$$\\end{document} norm of a strong solution u on the time interval [0, T] is bounded by A≫1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\gg 1$$\\end{document} then for each k≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 0 $$\\end{document} there exists Ck>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_k>1$$\\end{document} such that ‖Dku(t)‖L∞(R3)≤t-(1+k)/2expexpACk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert D^k u (t) \\Vert _{L^\\infty (\\mathbb {R}^3)} \\le t^{-(1+k)/2}\\exp \\exp A^{C_k}$$\\end{document} for all t∈(0,T]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in (0,T]$$\\end{document}.

Numerical Flowfield Visualization Over Heat Shield Of Satellite Launch Vehicle at Mach 0.8 - 3.0

The paper presents a numerical flowfield visualization over a typical payload shroud of a satellite launch vehicle at Mach number range of 0.80 -3.0 and Reynolds number range of 33 x 106 - 46 x 106 /m. The numerical simulation over the bulbous heat shield is carried out by solving time-dependent compressible axisymmetric turbulent Reynolds-averaged NavierStokes equations. The closure of these equations is obtained employing the Baldwin-Lomax turbulence model. A three-stage Runge-Kutta time-stepping scheme has been used in conjunction with finite-volume discretization of the computational domain. The flowfield features over the bulbous heat shield have been analyzed from the velocity vector and the density contour plots. Flow separation on the payload shroud due to the normal shock wave is observed at Mach number 0.80 and 0.90. The normal shock movement on the heat shield is simulated for various transonic Mach number. A recirculation zone of flowfield is formed in the boat-tail region of the heat shield. The vorticity formation ahead of the heat shield moves close to the heat shield with the increasing transonic Mach number, which is observed in the density contours plots of the flowfield. Shock stand-off distance at the supersonic speed is calculated and compared with asymptotic formula of Frank and Zierep. They are found in good agreement.

Flow Field Simulations Over Reentry Modules at High Speed

Reentry capsule configurations significantly differ from each other due to entry conditions and mission requirements. This paper describes numerical simulations of the viscous flow over the Beagle and the OREX (Orbital Reentry EXperiments) configurations for freestream Mach numbers in the range of 1.2 - 5.0. The flow fields over the reentry modules are obtained by solving time-dependent axisymmetric compressible laminar Navier-Stokes equations. The fluid mechanics equations are discretized in spatial coordinates employing a finite volume method, which reduces the governing equations to semi-discretized ordinary differential equations. Temporal integration is carried out using a two-stage Runge-Kutta time-stepping scheme. A local time-stepping is employed to get the steady state solution. The numerical simulation is performed on a single-block structured grid. The flow field features around the reentry capsules such as bow shock wave, sonic line, expansion fan and recirculation flow in the base-shell region are well captured by the present numerical computation. The effects of the geometrical parameters of the module, such as aspect ratio, frontal segment bluntness, fore body cone angle and shoulder rounding radius on the wake throat width, distance from the wake throat to the base of the model, flow departure angle and aerodynamic drag are analyzed using the numerically simulated flow field.

Numerical Simulation of Flow Field Over a Sharp-Tipped Double Cone at High Speed

The flowfield over a sharp-tipped double cone axisymmetric configuration is carried out by solving time dependent axisymmetric laminar compressible Navier-Stokes equations for freestream Mach number range of 2.0 - 6.0. The fluid dynamics equations are discretized in spatial coordinates using integral formulation in conjunction with finite volume method which reduces the governing equations to semi-discretized ordinary differential equations. Temporal integration is performed employing multistage Runge-Kutta time-stepping scheme. A local time step is used to achieve steady-state solution. The numerical computation is carried out on a mono-block with structured grids. The flowfield features over the sharp-tipped double cone configuration such as conical shock wave, separation region, separation and reattachment shock wave and bow shock wave in the cone region, expansion fan, and recirculation flow in the base region are well captured at Mach number 3 and 6 which are identical to the Edney VI type shock interaction. The velocity vector, Mach contours plots and variations of surface pressure coefficient along the sharp-tipped double cone configuration are analyzed at various Mach numbers. The fore-body aerodynamic drag is calculated employing computed pressure distribution. The paper presents the influence of the freestream Mach number on the flows with shock interactions over the sharp-tipped double cone geometry.

Aerothermodynamic Analysis of Space Recovery Experiment Capsule at Mach 6

The main focus of this paper is to compute the flowfield and assess the aerodynamic drag of Space Recovery Experiment (SRE) modules at freestream Mach 6 at zero angle of incidence. Time-dependent Compressible axisymmetric Navier-Stokes equations are numerically simulated using a finite volume discretization method with a three-stage Runge-Kutta time-stepping scheme. The computed shock stand-off distance, pressure ratio across normal shock and stagnation point heat flux are showing satisfactory results with the analytical solutions.Available wind tunnel data such as Schlieren images, oil flow pictures and fore-body aerodynamic drag are compared with numerical results.Influence of various ramp angles of the SRE modules on surface pressure, flow separation, skin friction coefficient, wall heat flux and fore and base aerodynamic drag coefficients are investigated using present numerical data.

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.

Dynamics of a Wavy Axisymmetric Swimmer

A novel underwater locomotion method is proposed. In this method, the propulsion system is designed as an empty tube with two open ends. The sidewall of this tube is deformable so that the diameter of the tube varies both spatially and temporally, following a wavy pattern that travels from head to tail. By using a fluid dynamics model based on the axisymmetric Navier–Stokes equations, we numerically demonstrate that this design is able to generate thrust for self-propelled swimming. The sources of the thrust have been identified. Specifically, the physical mechanisms that assist with the swimming performance are 1) the momentum flux through the front and back openings; and 2) the normal stress at the front and back openings, most notably the suction force at the front opening due to negative pressure there. Moreover, a parametric study has been conducted to investigate the effect of geometric and kinematic parameters upon the performance of the system.

Long-time asymptotics of axisymmetric Navier–Stokes equations in critical spaces

Long-time asymptotics of axisymmetric Navier–Stokes equations in critical spaces

On the secondary flows in oscillating-cup viscometer

Oscillating-cup viscometer is analysed numerically at high-amplitude regime. Full axi-symmetric Navier–Stokes equations are solved by a finite volume method. It is shown that the effect of the secondary flows on the decrement and period of the oscillations scales as square of the amplitude. The secondary flow consists of a stationary circulation and a periodic component whose period is a half of the period of the primary oscillations. A procedure allowing for determination of the linear decrement in a non-linear regime is proposed. It is shown that the constraints of “smallness” can be significantly relaxed and the experiments can be performed with higher amplitudes and higher decrements of the oscillations.

On the influence of non equilibrium in the free stream conditions of high enthalpy oxygen flows around a double-cone

This work presents a detailed analysis of the shock wave/boundary layer interaction in hypersonic flows around a double-cone. Numerical simulations have been carried out by solving the axisymmetric Navier–Stokes equations for an oxygen mixture. Thermochemical non-equilibrium is taken into account by employing the multi-temperature model (mT) proposed by Park and the State-to-State model (StS). The occurrence of surface processes is also analyzed considering both non-catalytic and fully-catalytic wall models. It is shown that the evaluation of correct free stream conditions is fundamental for the prediction of the separation extent and the corresponding heat transfer in high enthalpy conditions.

Potentially Singular Behavior of the 3D Navier–Stokes Equations

Whether the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of \(10^7\). We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya–Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of \(L_t^q L_x^p\) norm of the velocity with \(3/p + 2/q \le 1\). Our numerical results for the cases of \((p,q) = (4,8),\; (6,4),\; (9,3)\) and \((p,q)=(\infty ,2)\) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of \((p,q)=(3,\infty )\) is more difficult to verify numerically due to the extremely slow growth rate in the \(L^3\) norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global \(L^3\) norm of the velocity grows very slowly, the localized version of the \(L^3\) norm of the velocity experiences rapid dynamic growth relative to the localized \(L^3\) norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations.

Computational Fluid Dynamics Analysis and Design of Payload Shroud of Satellite Launch Vehicle

The main focus of the present paper is to computational fluid dynamics analysis and design of payload fairing of satellite launch vehicle at freestream Mach number range of 0.6 - 3.0. Initially, time-dependent compressible three-dimensional Euler equations are solved employing a finite volume discretization method with a multi-stage Runge-Kutta time-stepping scheme to compute surface pressure and aerodynamic coefficients at various payload fairing and at angle of attack up to 5o with an increment of 1o. Payload fairing dimensions are selected that satisfies permissible structure load on satellite launch vehicle. Detailed flowfield simulation is carried out on the selected payload fairing employing axisymmetric compressible Reynolds-average Navier-Stokes equations to assess unsteady flowfield characteristics. The numerical simulations are used to locate terminal shock on the payload fairing at transonic Mach number. Unsteady flow characteristics are used to compute acoustic load. Shock standoff distances at supersonic speeds are tabulated and compared with the analytical solution. Schlieren images and oil flow pictures are compared with experimental results and in good agreement. Aerodynamic shape optimization of satellite launch vehicle payload fairing shape has been performed to satisfy structural load at maximum drag and dynamic pressure.

One‐ and two‐step semi‐Lagrangian integrators for arbitrary Lagrangian–Eulerian‐finite element two‐phase flow simulations

AbstractSemi‐Lagrangian (SL) schemes are of utmost relevance to simulate two‐phase flows where advection dominates. The combination of SL schemes with the finite element (FE) method and arbitrary Lagrangian–Eulerian (ALE) dynamic meshes yields a strong ingredient to deal with applications in Earth sciences, environmental engineering, and oil and gas industry whose numerical background is incipient. This article is intended to propose a second‐order time multistep SL/ALE/FE scheme to track particle trajectories traveling amidst single‐ or two‐phase incompressible flows both in fixed and moving mesh situations. The novel scheme is a BDF2‐like implementation that considers the mesh velocity as part of its backward‐in‐time integration. Trajectory departure points are searched through extrapolation and followed by a multistep interpolation corrector that works both for constant and varying time steps. A series of two‐phase benchmark flow simulations are carried out by using the cubic element to compare the performance of the new integrator against single‐step approximations as well as to analyze enhancements in representing the two‐phase flow dynamics. We use the 2D axisymmetric Navier–Stokes equations as underlying model. Error analyses, convergence tests, quantitative, and qualitative comparisons are presented and discussed to highlight the superior conservative feature of the novel scheme.