Full Navier-Stokes Equations Research Articles

Topological asymptotic expansion for the full Navier–Stokes equations

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

The vibration of a spring damped vibrator under a long cylindrical squeeze film force

Squeeze film effects between two concentric cylinders are common and important in many mechanical systems. The models for cylindrical squeeze film force and vibrating system with squeeze film effects have not been well understood. In this paper, we model a cylindrical squeeze film force starting from the simplification of the full Navier–Stokes equations and then get an expression for the squeeze film force. We study the influence of a harmonic exciting force on a spring damped vibrator with squeeze film effect by establishing a non-linear mathematical model, and we investigate the behavior of the system numerically. As a forced vibrating system, the fundamental frequency of the response is consistent with the exciting frequency; the response of the resultant non-linear system shows obvious superharmonic phenomenon; the number of the superharmonic peaks increase with Reynolds number, and the superharmonic components mainly come from the non-linearity of the flow convection.

Full compressible Navier-Stokes equations with the Robin boundary condition on temperature

We consider full compressible Navier-Stokes equations with the Robin boundary condition on temperature. Note that the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature. It is shown that a unique global strong solution existed if the initial data belongs to H 1 . Subsequently, we find that the strong solution is nonlinearly exponentially stable as time tends to infinity. This result could be viewed as the first one on the global well-posedness of the strong solution to full Navier-Stokes equations in a bounded domain with the degenerate heat conductivity and the Robin boundary condition on temperature. The proofs are mainly based on the energy method and a special inequality.

Discrete unified gas kinetic scheme for continuum compressible flows.

In this paper, a discrete unified gas kinetic scheme (DUGKS) is proposed for continuum compressible gas flows based on the total energy kinetic model [Guo et al., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704]. The proposed DUGKS can be viewed as a special finite-volume lattice Boltzmann method for the compressible Navier-Stokes equationsin the double distribution function formulation, in which the mass and momentum transport are described by the kinetic equationfor a density distribution function (g), and the energy transport is described by the other one for an energy distribution function (h). To recover the full compressible Navier-Stokes equationsexactly, the corresponding equilibrium distribution functions g^{eq} and h^{eq} are expanded as Hermite polynomials up to third and second orders, respectively. The velocity spaces for the kinetic equationsare discretized according to the seventh and fifth Gauss-Hermite quadratures. Consequently, the computational efficiency of the present DUGKS can be much improved in comparison with previous versions using more discrete velocities required by the ninth Gauss-Hermite quadrature.

Prediction of the Absorption Characteristics of Non-Uniform Acoustic Absorbers with Grazing Flow

In this paper, planar and the cylindrical broadband non-uniform acoustic absorbers were constructed, both of which use broadband absorption units (BAUs) as their building blocks. The impedance boundary Navier–Stokes equation (IBNSE) method was developed to predict the absorption characteristics of the lined duct with non-uniform acoustic absorbers, in which each small piece of perforated plate is acoustically equivalent to a semi-empirical impedance model through the boundary condition. A total of four semi-empirical impedance models were compared under different control parameters. The full Navier–Stokes equation (FNSE) method was used to verify the accuracy of these impedance models. It was found that the IBNSE method with the Goodrich model had the highest prediction accuracy. Finally, the planar and the cylindrical non-uniform acoustic absorbers were constructed through spatial extensions of the BAU. The transmission losses and the absorption coefficients of the rectangular duct–planar acoustic absorber (RDPAA) and annular duct–cylindrical acoustic absorber (ADCAA) systems under grazing flow were predicted, respectively. The results demonstrated that the broadband absorption of the designed non-uniform acoustic absorbers was achieved. The developed IBNSE method with Goodrich model was accurate and computationally efficient, and can be used to predict the absorption characteristics of an acoustically treated duct in the presence of grazing flow.

Fluid Flow and Heat Transfer Characteristics of the Full-Scale Infrared Suppression Device With the Louvered Conical Funnels Considering Surface Radiation: A Three-Dimensional Numerical Analysis

Abstract The current analysis examines the thermofluid properties of an infrared suppression (IRS) system with louvered conical diathermic funnel walls in the presence of surface radiation. An IRS device's flow and heat transfer characteristics are challenging because cold atmospheric air is entrained through the interfunnel openings into the IRS and mixes hot exhaust gas from the turbine in naval/cargo ships. The full Navier–Stokes equation is solved along with the energy equation and radiative transfer equation (where applicable) for modeling a three-dimensional actual-scale IRS device with louvered conical funnel walls. This study elucidates the influence of the Reynolds number (6.1×105 to 3.18×106) nozzle overlapping, inclination angle, funnel overlapping, and guide vanes on the air intake and system exit temperature. The calculation of the lock-on range on the ship with or without the IRS device. The performance of a diathermic wall with surface radiation is significantly improved.

Modeling acoustic emissions and shock formation of cavitation bubbles

Despite significant progress in understanding and foretelling pressure-driven bubble dynamics, models that faithfully predict the emitted acoustic waves and the associated shock formation of oscillating or collapsing bubbles have received comparably little attention. We propose a numerical framework using a Lagrangian wave tracking approach to model the acoustic emissions of pressure-driven bubbles based on the Kirkwood–Bethe hypothesis and under the assumption of spherical symmetry. This modeling approach is agnostic to the equation of the state of the liquid and enables the accurate prediction of pressure and velocity in the vicinity of pressure-driven bubbles, including the formation and attenuation of shock fronts. We validate and test this new numerical framework by comparison with solutions of the full Navier–Stokes equations and by considering a laser-induced cavitation bubble as well as pressure-driven microbubbles in excitation regimes relevant to sonoluminescence and medical ultrasound, including different equations of state for the liquid. A detailed analysis of the bubble-induced flow field as a function of the radial coordinate r demonstrates that the flow velocity u is dominated by acoustic contributions during a strong bubble collapse and, hence, decays predominantly with u∝r−1, contrary to the frequently postulated decay with u∝r−2 in an incompressible fluid.

Low-order models for a circular hydraulic jump on a spherical cap

In this work, we analyze the flow of a thin layer of viscous liquid over the outer surface of a sphere due to inertia and gravity. We use the classical problem of a circular hydraulic jump as a starting point and observe the changes in the flow structure as the gravity component along the surface becomes significant. We assume that the flow is stationary and axisymmetric, the curvature of the spherical surface is small, and the capillary forces are negligible. The depth-averaged thin-layer equations describe the flow. We perform a qualitative analysis using a one-parametric representation of the longitudinal velocity distribution and find the necessary conditions for the hydraulic jump existence. The intensity of the jump monotonically decreases, and its radius grows to a certain finite value. The jump vanishes at a finite distance from the axis of symmetry. Using a two-parametric representation, we locate zones of recirculating flow and find the condition of their existence. We find the optimal strategy of averaging by comparing the results of our calculations with the data obtained experimentally and by using simulations in the framework of the full Navier–Stokes equations.

MODELING OF AERODYNAMIC INTERFERENCE IN THE DEVELOPMENT OF PERSPECTIVE AIRCRAFTS

There are presented the results of the main approaches analysis to modeling aerodynamic interference in the development of perspective aircrafts.
 The use of the methodological approaches proposed in the article will make it possible to determine the optimal aircraft layout in the formulation of the “wing-fuselage-engine” on the results of the achieved total aerodynamic characteristics.
 Analysis of modern armed conflicts shows that in most cases the use of aviation has a decisive influence on the outcome. Taking this into account, the main direction for the development of state aviation of Ukraine in the long term, with regard to the renewal of military transport aircraft fleet, is the designing and production of domestic planes. Nowadays, the world’s leading aircraft manufacturing companies (Airbus, Boeing and others) are working on the creation of competitive aircraft which are based on conceptually new schemes. This is because optimizing the shape of an aircraft can improve performance by only fraction of a percent while changes in the aircraft layout can lead to more significant positive results. Aerodynamic interference between the components of designed aircraft layout is relatively large and even minor changes in the geometry of one of the parts can significantly affect the overall aerodynamic characteristics of the aircraft. It is especially important to investigate the interference of working engines and airframe at different phases of flight at the stage of aircraft design.
 The purpose of the article is to present the results of the main approaches and methods analysis that can be used in modeling aerodynamic interference which is important in determining the total aerodynamic characteristics of advanced aircraft at the designing stage. This article uses the following research methods and methodologies:
 
 proven computational methods for optimizing the shape of an aircraft at the initial designing stage;
 boundary element methods, which are used to calculate potential flows, are also called methods of hydrodynamic singularities;
 numerical methods for studying the process of flow around bodies are used to study aero hydrodynamics based on various combinations of distributed and discrete hydrodynamic singularities;
 the numerical-analytical method and the panel method are used to calculate the carrier and moment characteristics of the airfoil in the framework of an ideal fluid;
 calculations based on the Navier-Stokes equations with a set of turbulence models are used for more complete and reliable information for calculating the carrier and moment characteristics of the airfoil;
 the discrete vortex method is used when the engine and the airframe are not considered separately from each other, but together;
 an iterative combination of the panel method with subsequent solution of the boundary layer equations for calculating the inviscid flow is used to take into account the effect of viscosity on the aerodynamic characteristics of the
 
 Conclusion: For solving problems of aerodynamic interference, the most suitable methods of the ideal fluid are the numerical-analytical and panel ones. More complete and reliable information on the calculation of the carrier and moment characteristics of the airfoil can be obtained by solving the full Navier-Stokes equations with a set of turbulence models. For those cases when it is necessary to obtain the aerodynamic characteristics of an aircraft that is designed with high accuracy.It is advisable to experimentally determine the corrections that are associated with the influence of operating engines.

Bifurcation phenomena in Taylor–Couette flow in a very short annulus with radial through-flow

In this study, the non-linear dynamics of Taylor–Couette flow in a very small-aspect-ratio wide-gap annulus in a counter-rotating regime under the influence of radial through-flow are investigated by solving its full three-dimensional Navier-Stokes equations. Depending on the intensity of the radial flow, either an axisymmetric (pure m=0 mode) pulsating flow structure or an axisymmetric axially propagating vortex will appear subcritical, i.e. below the centrifugal instability threshold of the circular Couette flow. We show that the propagating vortices can be stably existed in two separate parameter regions, which feature different underlying dynamics. Although in one regime, the flow appears only as a limit cycle solution upon which saddle-node-invariant-circle bifurcation occurs, but in the other regime, it shows more complex dynamics with richer Hopf bifurcation sequences. That is, by presence of incommensurate frequencies, it can be appeared as 1-, 2- and 3-torus solutions, which is known as the Ruelle–Takens–Newhouse route to chaos. Therefore, the observed bifurcation scenario is the Ruelle-Takens-Newhouse route to chaos and the period doubling bifurcation, which exhibit rich and complex dynamics.

Toward an asymptotic description of Prandtl–Batchelor flows with corners

The Prandtl–Batchelor theorem states that the vorticity in a steady laminar high Reynolds (Re) number flow containing closed streamlines should be constant; however, apart from the simple case of circular streamlines, very little is known about how to determine this constant (ω0). This paper revisits earlier work for flow driven by a surrounding smooth moving boundary, with a view to extending it to the case where the enclosing boundary has corners; for this purpose, a benchmark example from the literature for flow inside a semi-circle is considered. However, the subsequent asymptotic analysis for Re≫1 and numerical experimentation lead to an inconsistency: the asymptotic approach predicts boundary-layer separation, whereas a linearized asymptotic theory and computations of the full Navier–Stokes equations for Re≫1 do not. Nevertheless, by considering a slightly modified problem instead, which does not suffer from this inconsistency, it is found that, when extrapolating the results of such high-Re computations to infinite Re, the agreement for ω0 is around 5%, which is roughly in line with previous comparisons of this type. Possible future improvements of the asymptotic method are also discussed.

Energy conservation law for weak solutions of the full compressible Navier-Stokes equations

We consider a sufficient condition for the energy conservation law of a weak solution for the full compressible Navier-Stokes equations on the torus. We prove that a weak solution constructed by Feireisl with certain integrability conditions conserves the energy. Our assumption relaxes the regularity condition for the density compared with existing results.

Simulation of Micro-Powder Injection Molding Using a Dynamic Mesh Approach

Abstract Powder injection molding (PIM) is capable of manufacturing complex small components out of metals and ceramics. With increasing market demand micropowder injection molding (micro-PIM) is improving rapidly which calls for improved numerical models of different stages of PIM. The injection stage is quite important as different defects may occur during injection that cannot be eliminated in the subsequent processes. Injection pressure and velocity are critical controlling parameters to manufacture a defect-free component. However, in most of the simulations, either injection pressure or injection velocity is taken as a constant value, while in reality, the injection molding of the piston moves at different speeds, and injection pressure and velocity change accordingly. In the experimental part of the present work, the piston movement is tracked using a high-speed camera which was mimicked in mold filling simulation of an alumina feedstock, considering the Dynamic Mesh approach. The solution of the full Navier–Stokes equation in an Eulerian-multiphase framework is used to simulate the mold filling where the secondary phase is air that is displaced by moving melt front. Additionally, a scalar equation was used to account for the liquid fraction of the feedstock. It is further demonstrated that the proposed model provides insight into the fluid-dynamic aspect of the mold-filling process.

Global Well-Posedness for the Full Compressible Navier-Stokes Equations

We are concerned with the Cauchy problem regarding the full compressible Navier-Stokes equations in ℝd (d = 2, 3). By exploiting the intrinsic structure of the equations and using harmonic analysis tools (especially the Littlewood-Paley theory), we prove the global solutions to this system with small initial data restricted in the Sobolev spaces. Moreover, the initial temperature may vanish at infinity.

Thermo-Hydraulic Performance of Pin-Fins in Wavy and Straight Configurations.

Pin-fins configurations have been investigated recently for different engineering applications and, in particular, for a cooling turbine. In the present study, we investigated the performance of three different pin-fins configurations: pin-fins forming a wavy mini-channel, pin-fins forming a straight mini-channel, and a mini-channel without pin-fins considering water as the working fluid. The full Navier–Stokes equations and the energy equation are solved numerically using the finite element technique. Different flow rates are studied, represented by the Reynolds number in the laminar flow regime. The thermo-hydraulic performance of the three configurations is determined by examining the Nusselt number, the pressure drop, and the performance evaluation criterion. Results revealed that pin-fins forming a wavy mini-channel exhibited the highest Nusselt number, the lowest pressure drop, and the highest performance evaluation criterion. This finding is valid for any Reynolds number under investigation.

Low Mach number limit of Navier-Stokes equations with large temperature variations in bounded domains

The low Mach number limit of full compressible Navier-Stokes equations with large temperature variations is verified rigorously in a three-dimensional bounded domain. Weighted uniform estimates of the solutions are derived in a time interval which is independent of the Mach number, in particular, for the high-order derivatives, when the initial data are well-prepared only in the sense of L2-norm. It can be viewed as the first result on the low Mach number limit of full Navier-Stokes equations with large temperature variations in bounded domains. The methods in the paper also apply to other fluid models with large temperature variations.

Thermohydraulic Performance of Chevron Pin-Fins

The present study focuses on the optimum design effectiveness in heat removal for small surfaces. Pin-fin made of solid and porous cylindrical shape forming chevron is investigated numerically using the finite element method. The design consists of 3-chevron and 5-chevron configurations connected to a heated block with fluid circulating between the chevron and above them. Variable Reynolds number and pin-fins height ranging from 2 mm to 8 mm are investigated. The full Navier–Stokes equation combined with the energy equation was solved in the presence of the solid pin-fins. The Darcy–Brinkman model with the effective energy equation is used in the presence of the porous pin-fins. The system is solved for Reynolds numbers ranging from 50 to 1000, thus remaining in the laminar regime. Results revealed that the best performance evaluation criterion is higher for the 8 mm porous pin-fins regardless of their permeability. If one ignores the pressure drop and friction contribution, a solid pin-fin having a height of 4 mm showed the best heat absorption mechanism.

Heat Transfer Due to Annular Jets Impinging on a Moving Surface

Abstract This work investigates flow and heat transfer under an array of annular jets impinging on a heated moving surface. Numerical solutions of the full Navier–Stokes equation were attempted with a highly refined mesh. This study reports results for Reynolds numbers up to 500. In the surface movement direction, a periodic element from a jet-bank configuration was chosen, and the nondimensional surface velocity was considered from zero (i.e., a stationary plate) to two times the jet velocity. The impact of annular jet impingement over a moving surface on flow and heat transfer characteristics, including the development of the flow field, velocity profiles, skin friction coefficient and topology of skin friction lines, and local as well as surface averaged Nusselt number distribution are presented. It is observed that both the flow field and thermal performance are strongly affected by the surface motion. Heat transfer from the surface initially increases with the increasing surface motion, and after attainment of the highest value, heat transfer reduces with a further increase in surface velocity. However, higher surface velocity leads to higher uniformity in heat transfer, which may be beneficial for situations demanding uniformity in heat transfer.

Thermo-fluid characteristics of an IRS system with louvered cylindrical diathermic funnels considering surface radiation: A three-dimensional numerical exercise

The thermo-fluid characteristics of an IRS (infrared suppression) system with louvered cylindrical diathermic funnels considering surface radiation are investigated in the present analysis. An IRS device's flow and heat transfer features are challenging as the cold atmospheric air is entrained into the system through the inter-funnel openings and mixed with hot exhaust gas from the turbine in cargo/naval ships. The numerical simulations of the full Navier–Stokes equation with the energy equation and radiative transfer equation (when required) are carried out for the three-dimensional IRS model with louvered cylindrical funnels. This study elucidated the influence of louver position, louver pattern, Reynolds number (6.1 × 105 to 3.18 × 106), funnel overlapping (zero, negative, and positive), nozzle overlapping, and nozzle shape (circular, square, triangular, and lobed) on mass intake and system exit temperature in detail. Zero overlapping cases or/and lobed shape of the nozzle are better than other considered cases. For the funnel walls, three different boundary conditions, namely adiabatic wall, diathermic wall, and diathermic wall with surface radiation, are also considered for the present analysis. The diathermic wall with surface radiation conditions was found to improve the performance appreciably.