Dimensional Navier-Stokes Equations Research Articles

Pore-scale resolved simulation of quenching, acceleration, and transition to detonation of hydrogen explosions by metal foams

Pore-scale simulations of quenching, acceleration, and transition to detonation of gaseous hydrogen explosions by metal foams were performed. The model solves the three dimensional compressible reactive Navier-Stokes equations with detailed chemistry in a closed square channel with a metal foam placed inside. The structures of the foam were fully resolved using an immersed boundary method. Stoichiometric mixtures of hydrogen-oxygen and hydrogen-air were considered. An explosion in the channel was initiated with a high-pressure and high-temperature spark. The simulation results show that the explosion can be quenched, accelerate, or transition to detonation depending on the foam pores-per-inch (PPI). Hydrogen-air explosions were quenched with foams of ∼39 PPI or higher due to heat losses from the flame, which is consistent with experimental observations. The explosion was not quenched by foams with low PPI. Instead the low PPI foams exacerbated the explosion by causing flame acceleration. Worse, a 78 PPI foam caused a stoichiometric H2 and O2 mixture to transition to detonation. Hydraulic resistance due to products venting away from the flame surface was found to produce strong pressure gradients within the foam. This causes a self-pressurizing effect that grows as the flame propagates further into the foam. The pressure within the foam grows and steepens into a shock. The shock initiated the detonation by a Mach reflection, which is similar to deflagration-to-detonation transition mechanisms in traditional obstacle-laden channels. Statement of SignificanceMetal foam structures are often used as devices to inhibit flames in scenarios where explosion hazards are a concern. Direct numerical simulations were performed to increase understanding of how flames and explosions interact with metal foams. The simulation results show that high pores-per-inch (PPI) foams quench the explosion by removing heat from the flame. Lower PPI foams can have the opposite effect and exacerbate the explosion by promoting flame acceleration or even transition to detonation. Flames self-pressurize as they propagate through low PPI foams due to hydrodynamic friction and gas-dynamic effects from flowing product gases. In extreme cases, the pressure waves near the flame can steepen into a shock and initiate a detonation. These results indicate that improperly selected metal foam flame arresters can exacerbate the explosion.

CFD study of a horizontal axis wind turbine NREL Phase II

The present work aims to study the aerodynamic characteristics of the NREL phase II (generated only with S809 profile along the span for an untwisted case) rotor that is a horizontal axis downwind wind turbine rotor and which is assumed to stand isolated in the space. The two dimensional steady-incompressible flow Reynolds average NavierStokes equations, are solved by using the commercial CFD package Ansys Fluent. The 2D computations are first performed on S809 airfoil in order to define the most suitable model to be used; the turbulence closure model has been chosen among four possible candidates (standard , Spalart-Allmaras, based on comparison of pressure coefficient for the different configurations with experimental results. Secondly, through a three dimensional study we tried to simulate the experiment for wind speed velocities of 7.2, 10.56, 12.85, 16.3, and 9.18m/s. Results of pressure and torque for considered wind turbine rotor have been directly compared to the available experimental data. The comparisons show that CFD results along with the turbulence model used can predict the span-wise loading of the wind turbine rotor with reasonable agreement. The work presented here is the first stage of project that aims at giving a better understanding of the main influence of the rotational effect on boundary layer separation, and identify the stalled configuration in order to control this latter in future work.

Thermochemical Exploration of a Cavity Based Supersonic Combustor with Liquid Kerosene Fuel

Thermochemical exploration of a liquid hydrocarbon fueled scram jet combustor is presented. Three dimensional Navier Stokes equations along with K-ε turbulence model and single step kerosene-air reaction kinetics are solved using commercial software. Various combustor configurations with different fuel injection cavities are analyzed. Simulations capture all the essential features of the flow field. Good comparisons between computational and experimental surface pressure form the basis for further analysis. Parametric studies have been carried out with different droplet diameters to study its effect in the flow development. The numerical simulation also confirmed the experimental observation that the threshold value of length-to-depth ratio for cavity characterization is different for reacting and non-reacting flows.

Performance Improvement of Kerosene Fuelled Scramjet Combustor Through Modified Fuel Injection - A Cfd Study

Full scale kerosene fuelled scramjet combustor of a hypersonic air-breathing vehicle is numerically explored for two different fuel injection systems. A commercial software CFXTASCflow has been used for the reacting flow computation within the combustor. Three dimensional Navier-Stokes equations are solved along with k-ε turbulence model to analyse the flow field. Eddy Dissipation Model (EDM) with fast rate chemical kinetics is used for combustion modeling. Lagrangian Particle Tracking Method (LPTM) has been used to simulate the trajectory of the kerosene droplets. Kerosene is injected from the struts. Reacting flow simulations have been carried out for the fuel equivalence ratio of 1.0. The overall flow characteristics in the combustor are presented through the distribution of various thermochemical parameters at different cross sections. Kerosene which is injected in liquid phase completely vaporizes within the combustor. Considerable improvement in the performance in terms of thrust and combustion efficiency is observed with the modified fuel injection system.

Numerical Investigation of Confinement Effect on Supersonic Turbulent Flow Past Backward Facing Step With And Without Transverse Injection

Effect of confinement has been investigated numerically for supersonic turbulent flow past backward facing step in a nonreacting scramjet combustor. Flow structure downstream of the backward facing step has been studied by considering various configurations with different combustor heights as well as unconfined flow. Staged transverse sonic injectors with different combustor heights are also considered to find out the effect of confinement on the penetration, spreading and other flow features in the flow field. Three dimensional Navier Stokes equations along with k-ε turbulence model are solved using a commercial CFD software. The simulation captures all essential features of the flow. Good comparisons of various flow profiles have been obtained between experimental and computed values. Although confinement creates complicated shock reflections in the combustor, the length of the recirculation bubble behind the backward facing step remains almost constant. For the injection case, the recirculation region is extended upto the bow shock arising upstream of the injector and two recirculation bubble is seen between the backward facing step and the injector.

CFD Prediction of Ramjet Intake Characteristics at Angle of Attack

The characteristics of installed intakes of a ramjet engine at angle of attack are predicted numerically. Three dimensional Navier Stokes equations are solved alongwith k - ε turbulence model using commercial CFD software. The software was first validated for isolated intake and computed results match very well with theoretical results available in literature. Pressure recovery Vs. mass flow characteristics of four installed intakes placed in a rear location of a ramjet vehicle is evaluated at different angles of attack upto 6°. Intake performance matches reasonably well with experimental results. It has been observed that significant variation of intake performance exists between windward and leeward intakes. For higher angle of attack, the leeward intake move towards the subcritical operation faster and the intake flowfield is seen to interact with the corebody boundary layer causing significant spillage.

Numerical Simulation of Underexpanded Sonic Jet

Underexpanded sonic jet is numerically simulated using commercial CFD software. Three dimensional Navier Stokes equations alongwith two equation turbulence models are solved. Computed centerline distribution and profiles of various flow variables at various axial stations agree quantitatively with experimental results. The comparison of the flow variables computed with four different two equation models reveals that k - e and RNG k - e models perform very well in predicting both pressure and temperature throughout the flow filed; whereas SST and k - w models fail to predict the flow behavior beyond the first Mach disc.

Numerical ergodicity of two dimensional stochastic Navier-Stokes equations with Gaussian noise

Numerical ergodicity of two dimensional stochastic Navier-Stokes equations with Gaussian noise

Time-fractional (2+1)-dimensional navier-stokes equations: similarity reduction and exact solutions for one-parameter lie group of rotations

The current study is dedicated to solving the time-fractional (2+1)-dimensional Navier–Stokes model. The model has wide applications in blood flow, in the design of power stations, weather prediction, ocean currents, water flow in a pipe, air flow around the aircraft wings, the analysis of pollution, and many other areas of engineering. The Lie symmetry approach is applied to the governed time-fractional equation to fulfill this need. In the direction of exact solutions of the time-fractional equation first of all invariance condition is obtained in the presence of the Lie group. Consequently, infinitesimals are obtained with the help of the invariant condition. Moreover, these infinitesimals are utilized to obtain the subalgebras. Further, under each subalgebras similarity variables and similarity solutions are obtained which are used to find the reduced equations. These reduced equations are solved for exact solutions. The solutions of the reduced equations are further used to find the exact solutions of the main time-fractional (2+1)-dimensional Navier–Stokes equation with the help of similarity solutions under each subalgebra.

Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case

We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on $${\mathbb {T}}\times {\mathbb {R}}$$ . The main task is to understand the associated Rayleigh and Orr–Sommerfeld equations, under the natural assumption that the linearized operator around the monotonic shear flow in the inviscid case has no discrete eigenvalues. We obtain precise control of solutions to the Orr–Sommerfeld equations in the high Reynolds number limit, using the perspective that the nonlocal term can be viewed as a compact perturbation with respect to the main part that includes the small diffusion term. As a corollary, we give a detailed description of the linearized flow in Gevrey spaces (linear inviscid damping) that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to accurately capture the behavior of the solution to Orr–Sommerfeld equations in the critical layer. In this paper we consider the case of shear flows on $${\mathbb {T}}\times {\mathbb {R}}$$ . The case of bounded channels poses significant additional difficulties, due to the presence of boundary layers, and will be addressed elsewhere.

Pressure, Intermittency, Singularity

We give conditions for regularity of solutions of three dimensional incompressible Navier-Stokes equations based on the pressure and on structure functions.

Global existence of strong solutions to the kinetic Cucker-Smale model coupled with the two dimensional incompressible Navier-Stokes equations

<p style='text-indent:20px;'>In this paper, we investigate existence of global-in-time strong solutions to the Cauchy problem of the kinetic Cucker–Smale model coupled with the incompressible Navier–Stokes equations in the two dimensional space. By introducing a weighted Sobolev space and using the maximal regularity estimate on the linear non-stationary Stokes equations, we present a complete analysis on existence of global-in-time strong solutions to the coupled model, without any smallness assumptions on initial data.</p>

Local $ \varepsilon $-regularity criteria for the five dimensional stationary Navier-Stokes equations

We establish interior and boundary $ \varepsilon $-regularity criteria at one scale for suitable weak solutions to the five dimensional stationary incompressible Navier-Stokes equations which improve previous results in [19] and [32]. Our proof is based on an iteration argument, Campanato's method, and interpolation techniques.

Large global solutions of the compressible Navier-Stokes equations in three dimensions

<p style='text-indent:20px;'>This work concerns the initial value problem for the three dimensional compressible Navier-Stokes equations (both isentropic and polytropic). By exploiting the famous Fujita-Kato theorem to the Classical incompressible Navier-Stokes equations, we prove the existence of global-in-time unique solutions under as weak as possible smallness conditions in the scaling invariant spaces. In particular, our results improve the classical theorems obtained by Danchin [Invent. Math., <b>141</b>, 579–614, 2000] and Danchin [Arch. Ration. Mech. Anal., <b>160</b>, 1–39, 2001].</p>

A comprehensive study of the onset of boundary layer separation in the unbounded flow around a circular cylinder

The onset of boundary layer separation in the unbounded flow around a circular cylinder is determined directly with a numerical experiment with accuracy to the second decimal point of the Reynolds number, Re. The governing equations are the steady state, two dimensional Navier–Stokes equations, which are solved with Galerkin finite elements. A new criterion, the entrance ratio E=D/L with D the diameter of the cylinder and L the entrance length of the domain, is introduced in addition to the traditional blockage ratio; the aim is to establish conditions for unbounded flow in both flow directions. A hypothesis is formulated and verified for the onset of separation leading to the conclusion that this flow undergoes a supercritical pitchfork bifurcation with respect to fixed points on the boundary. Results are presented for various blockage ratios, B, in the range 1.2238·10−8≤B≤0.005 and constant entrance ratio E=2.4472·10−9 until asymptotic behavior is observed for the onset of separation in the Reynolds number, in the angle and the length of the smallest eddy. Onset of separation occurs in the region 6.36≤Re<6.37. The bifurcation diagram of the flow is calculated. Streamlines of the flow domain before and after the bifurcation point are shown, including images of the smallest possible eddies. In addition, coefficients of base suction and total drag are computed and shown in the regime 0.1≤Re≤40. The results are compared with available laboratory and numerical measurements.

How to Extract a Spectrum from Hydrodynamic Equations

The practical results gained from statistical theories of turbulence usually appear in the form of an inertial range energy spectrum $${\mathcal {E}}(k)\sim k^{-q}$$ and a cutoff wavenumber $$k_{c}$$ . For example, the values $$q=5/3$$ and $$\ell k_{c}\sim \mathrm {Re}^{3/4}$$ are intimately associated with Kolmogorov’s 1941 theory. To extract such spectral information from the Navier–Stokes equations, Doering and Gibbon (Phys. D 165, 163–175, 2020) introduced the idea of forming a set of dynamic wavenumbers $$\kappa _n(t)$$ from ratios of norms of solutions. The time averages of the $$\kappa _n(t)$$ can be interpreted as the 2nth moments of the energy spectrum. They found that $$1 < q \leqslant 8/3$$ , thereby confirming the earlier work of Sulem and Frisch (J. Fluid Mech. 72, 417–423, 1975) who showed that when spatial intermittency is included, no inertial range can exist in the limit of vanishing viscosity unless $$q \leqslant 8/3$$ . Since the $$\kappa _n(t)$$ are based on Navier–Stokes weak solutions, this approach connects empirical predictions of the energy spectrum with the mathematical analysis of the Navier–Stokes equations. This method is developed to show how it can be applied to many hydrodynamic models such as the two dimensional Navier–Stokes equations (in both the direct- and inverse-cascade regimes), the forced Burgers equation and shell models.

On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles

In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension $d\ge 2$ with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated acoustic cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic $\mathcal{C}^\infty$ self-similar solutions with suitable decay at infinity. The $\mathcal{C}^\infty$ regularity is used in a fundamental way in our companion paper (part II) in the analysis of the associated linearized operator and leads, in turn, to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions $d=2,3$.

Angiodynamic and optical coupling analysis of skin tissue model under finite pressure

The pulse amplitude of fingertip volume could be improved by selecting the vascular dense area and applying appropriate pressure above it. In view of this phenomenon, this paper used Comsol Multiphysics 5.6 (Comsol, Sweden), the finite element analysis software of multi-physical field coupling simulation, to establish the vascular tissue model of a single small artery in fingertips for simulation. Three dimensional Navier-Stokes equations were solved by finite element method, the velocity field and pressure distribution of blood were calculated, and the deformation of blood vessels and surrounding tissues was analyzed. Based on Lambert Beer's Law, the influence of the longitudinal compression displacement of the lateral light surface region and the tissue model on the light intensity signal is investigated. The results show that the light intensity signal amplitude could be increased and its peak value could be reduced by selecting the area with dense blood vessels. Applying deep pressure to the tissue increased the amplitude and peak of the signal. It is expected that the simulation results combined with the previous experimental experience could provide a feasible scheme for improving the quality of finger volume pulse signal.

Partially regular weak solutions of the stationary Navier-Stokes equations in dimension 6

By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force f in L_{text {loc}}^q cap L^{3/2}, q>3 in general open subdomains of {mathbb {R}}^6. These weak solutions satisfy certain local energy estimates and we estimate the size of their singular sets in terms of Hausdorff measures. We also prove the defect measures vanish under a smallness condition, in contrast to the nonstationary Navier-Stokes equations in {mathbb {R}}^4 times [0,infty [.

Fully decoupled, linear and unconditional stability implicit/explicit scheme for the natural convection problem

The aim of this paper is to develop an unconditionally stable Euler implicit/explicit scheme for the natural convection equations based on an introduced exponential scalar auxiliary variable. Our numerical scheme involves treating the linear terms in the implicit scheme and nonlinear terms in the explicit scheme. Then, the considered problem is decoupled into a series of linearized subproblems, and these algebraic systems with constant coefficient matrixes can be solved efficiently. Compared with the existing theoretical findings of the implicit/explicit scheme for the incompressible flow [He YN. The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or nonsmooth initial data. Math Comput. 2008;77(264):2097–2124.; He YN, Huang PZ, Feng XL. H2-stability of first order fully discrete schemes for the time-dependent Navier-Stokes equations. J Sci Comput. 2015;62:230–264; He YN, Li J. A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J Comput Appl Math. 2010;235(3):708–725; Jin JJ, Zhang T, Li J. H2 stability of the first order Galerkin method for the Boussinesq equations with smooth and nonsmooth initial data. Comput Math Appl. 2018;75(1):248–288; Su J, He YN. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete Continuous Dyn Syst Ser B. 2017;22(9):3421–3438], our analysis removes the restriction on time step , and obtains the unconditional stability and convergence theoretical findings. Some numerical results are also provided to verify the established theoretical analysis and show the performances of the considered numerical scheme.