Incompressibility Condition Research Articles

Alternative kinetic theory based lattice Boltzmann model for incompressible axisymmetric flows

Based on the axisymmetric Boltzmann equation, an incompressible lattice Boltzmann model for axisymmetric flows is proposed within the framework of the kinetic theory based model developed by Guo et al. (2009). While retaining the advantages of the Guo et al. (2009) model in terms of the solid physics basis and simple source terms involving no gradient calculations, the present model further improves the numerical stability, and reduces compressibility errors and computational requirements. Armed with the assumption that the fluid density is a constant and thus the fluid pressure has no direct relation with the density, the incompressibility conditions are realized by applying the Hermite expansion. Then, the present model employs a novel way of calculating the fluid pressure which is derived from the modified second-order moment equation. Additionally, based on the regularized lattice BGK (RLBGK) model, an extra relaxation parameter pertaining to the ghost mode is introduced to enhance the numerical stability of the present model. The accuracy and applicability of the present model are verified by both the Chapman–Enskog theoretical analysis and numerical validations. It is demonstrated via well-acknowledged test cases that the present model is accurate and reliable for incompressible axisymmetric flows, and is able to effectively reduce the compressibility errors vis-à-vis the Guo et al. (2009).

Linearized Euler Equations for the Determination of Scattering Matrices for Orifice and Perforated Plate Configurations in the High Mach Number Regime

The interaction of a plane acoustic wave and a sheared flow is numerically investigated for simple orifice and perforated plate configurations in an isolated, non-resonant environment for Mach numbers up to choked conditions in the holes. Analytical derivations found in the literature are not valid in this regime due to restrictions to low Mach numbers and incompressible conditions. To allow for a systematic and detailed parameter study, a low-cost hybrid Computational Fluid Dynamic/Computational Aeroacoustic (CFD/CAA) methodology is used. For the CFD simulations, a standard k–ϵ Reynolds-Averaged Navier–Stokes (RANS) model is employed, while the CAA simulations are based on frequency space transformed linearized Euler equations (LEE), which are discretized in a stabilized Finite Element method. Simulation times in the order of seconds per frequency allow for a detailed parameter study. From the application of the Multi Microphone Method together with the two-source location procedure, acoustic scattering matrices are calculated and compared to experimental findings showing very good agreement. The scattering properties are presented in the form of scattering matrices for a frequency range of 500–1500 Hz.

Modeling of Hydrophobic Surfaces by the Stokes Problem With the Stick–Slip Boundary Conditions

Unlike the Navier boundary condition, this paper deals with the case when the slip of a fluid along the wall may occur only when the shear stress attains certain bound which is given a priori and does not depend on the solution itself. The mathematical model of the velocity–pressure formulation with this type of threshold slip boundary condition is given by the so-called variational inequality of the second kind. For its discretization, we use P1-bubble/P1 mixed finite elements. The resulting algebraic problem leads to the minimization of a nondifferentiable energy function subject to linear equality constraints representing the discrete impermeability and incompressibility condition. To release the former one and to regularize the nonsmooth term characterizing the stick–slip behavior of the algebraic formulation, two additional vectors of Lagrange multipliers are introduced. Further, the velocity vector is eliminated, and the resulting minimization problem for a quadratic function depending on the dual variables (the discrete pressure and the normal and shear stress) is solved by the interior point type method which is briefly described. To justify the threshold model and to illustrate the efficiency of the proposed approach, three physically realistic problems are solved and the results are compared with the ones solving the Stokes problem with the Navier boundary condition.

Wave energy harvesting using a novel turbine rotor geometry

Wells turbines provide a practical solution for wave energy harvesting. The low aerodynamic efficiency of Wells turbines tangibly reduces their output power. Both the turbine efficiency and output power depend on the turbine solidity. The turbine solidity decreases from rotor hub to rotor tip for the commonly used rotors with constant chord-length blades. The present work introduces a novel Wells turbine rotor geometry. This geometry was obtained by numerically optimizing the rotor's radial solidity distribution. The turbine performance with different rotor geometries was numerically simulated by solving the three-dimensional Reynolds-averaged Navier–Stocks equation under incompressible and steady state flow conditions. Simple and multi-objective optimization were implemented in order to obtain the optimum rotor geometry. The present work showed that an improved turbine performance can be achieved by optimizing the turbine radial solidity distribution. Two different optimized rotor geometries were obtained and presented. The first rotor geometry improved the turbine efficiency by up to 4.7% by reducing its pressure drop. The second rotor geometries enhanced the turbine output power by up to 10.8%. Copyright © 2016 John Wiley & Sons, Ltd.

A short note on Navier–Stokes flows with an incompressible interface and its approximations

In biological applications, a cell membrane consisting of a lipid bilayer usually behaves as fluid-like interface with surface incompressibility. Here we consider a mathematical formulation for an incompressible interface immersed in Navier–Stokes flows and study the mathematical and physical features for this incompressible interface. The model formulation introduces an unknown tension which acts as a Lagrange’s multiplier to enforce such surface incompressibility. In this note, we show that the spreading operator of the tension and the surface divergence operator of the velocity are skew-adjoint with each other which indicates physically that the tension does not do extra work to the fluid under the condition of surface incompressibility. In order to avoid solving the unknown tension to enforce the surface incompressibility, we adopt a nearly surface incompressible approach (or penalty approach) by introducing two different modified elastic tensions which can be used efficiently in practical numerical simulations. Furthermore, we show that the resultant modified elastic forces have the same mathematical form as the original one derived from the unknown tension.

A truly incompressible smoothed particle hydrodynamics based on artificial compressibility method

In the present study, a truly incompressible smoothed particle hydrodynamics based on the artificial compressibility method for simulating steady and unsteady incompressible flows is proposed and assessed. The incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method proposed by Chorin in the Eulerian reference frame are written in a Lagrangian reference frame to provide an appropriate incompressible SPH algorithm. The proposed SPH formulation implemented here is based on an implicit dual-time stepping scheme to be capable of time-accurate analysis of unsteady flows. The advantage of the Artificial Compressibility-based Incompressible SPH (ACISPH) method proposed over the Weakly Compressible SPH (WCSPH) and the pseudo-compressibility SPH methods is that the ACISPH formulation is a truly incompressible SPH algorithm and it does not involve any approximate enforcement of the incompressibility condition that usually causes to use a large magnitude speed of sound implying a small time step in the computations. The approximate enforcement of the incompressibility condition used in the WCSPH method also causes spurious oscillations in the density and pressure fields which is not the case for the ACISPH formulation proposed. Unlike the projection SPH algorithm and the moving-particle semi-implicit (MPS) method formulation, the ACISPH formulation presented herein does not involve the iterative solution of a Poisson equation for the pressure field. The accuracy and robustness of the proposed incompressible SPH (ACISPH) are demonstrated by solving different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of particle resolution and the value of artificial compressibility parameter on the accuracy and convergence rate of the solution. To validate the results, different test cases are considered herein that are incompressible flows in a 2-D simple channel, a 2-D locally expanded channel, a 2-D cavity, the unsteady Couette flow with pressure gradient, the Taylor vortex problem and a dam break problem with dry bed. Results obtained for these cases are in good agreement with the available analytical and numerical results. The study shows the artificial compressibility-based ISPH (ACISPH) method proposed is accurate and robust for simulating steady and unsteady incompressible flow problems.

Measurements and Modeling of Ingress in a New 1.5-Stage Turbine Research Facility

In gas turbines, hot mainstream flow can be ingested into the wheel-space formed between stator and rotor disks as a result of the circumferential pressure asymmetry in the annulus; this ingress can significantly affect the operating life, performance, and integrity of highly stressed, vulnerable engine components. Rim seals, fitted at the periphery of the disks, are used to minimize ingress and therefore reduce the amount of purge flow required to seal the wheel-space and cool the disks. This paper presents experimental results from a new 1.5-stage test facility designed to investigate ingress into the wheel-spaces upstream and downstream of a rotor disk. The fluid-dynamically scaled rig operates at incompressible flow conditions, far removed from the harsh environment of the engine which is not conducive to experimental measurements. The test facility features interchangeable rim-seal components, offering significant flexibility and expediency in terms of data collection over a wide range of sealing flow rates. The rig was specifically designed to enable an efficient method of ranking and quantifying the performance of generic and engine-specific seal geometries. The radial variation of CO2 gas concentration, pressure, and swirl is measured to explore, for the first time, the flow structure in both the upstream and downstream wheel-spaces. The measurements show that the concentration in the core is equal to that on the stator walls and that both distributions are virtually invariant with radius. These measurements confirm that mixing between ingress and egress is essentially complete immediately after the ingested fluid enters the wheel-space and that the fluid from the boundary layer on the stator is the source of that in the core. The swirl in the core is shown to determine the radial distribution of pressure in the wheel-space. The performance of a double radial-clearance seal is evaluated in terms of the variation of effectiveness with sealing flow rate for both the upstream and the downstream wheel-spaces and is found to be independent of rotational Reynolds number. A simple theoretical orifice model was fitted to the experimental data showing good agreement between theory and experiment for all cases. This observation is of great significance as it demonstrates that the theoretical model can accurately predict ingress even when it is driven by the complex unsteady pressure field in the annulus upstream and downstream of the rotor. The combination of the theoretical model and the new test rig with its flexibility and capability for detailed measurements provides a powerful tool for the engine rim-seal designer.

Calculation of eigenvalues of Sturm-Liouville equation for simulating hydrodynamic soliton generated by a piston wave maker.

This paper focuses on the mathematical study of the existence of solitary gravity waves (solitons) and their characteristics (amplitude, velocity, ldots ) generated by a piston wave maker lying upstream of a horizontal channel. The mathematical model requires both incompressibility condition, irrotational flow of no viscous fluid and Lagrange coordinates. By using both the inverse scattering method and a given initial potential f_{0}(r), we can transform the KdV equation into Sturm–Liouville spectral problem. The latter problem amounts to find negative discrete eigenvalues lambda and associated eigenfunctions psi , where each calculated eigenvalue lambda gives a soliton and the profile of the free surface. For solving this problem, we can use the Runge–Kutta method. For illustration, two examples of the wave maker movement are proposed. The numerical simulations show that the perturbation of wave maker with hyperbolic tangent displacement under physical conditions affect the number of solitons emitted.

Efficient methods for solving the Stokes problem with slip boundary conditions

The paper deals with the Stokes flow with the threshold slip boundary conditions. A finite element approximation of the problem leads to the minimization of a non-differentiable energy functional subject to two linear equality constraints: the impermeability condition on the slip part of the boundary and the incompressibility of the fluid. Eliminating the velocity components, one gets the smooth dual functional in terms of three Lagrange multipliers. The first Lagrange multiplier regularizes the problem. Its components are subject to simple bounds. The other two Lagrange multipliers treat the impermeability and the incompressibility conditions. The last Lagrange multiplier represents the pressure in the whole domain. The solution to the dual problem is computed by an active set strategy and a path-following variant of the interior-point method. Numerical experiments illustrate computational efficiency.

Consistent lattice Boltzmann methods for incompressible axisymmetric flows

In this work, consistent lattice Boltzmann (LB) methods for incompressible axisymmetric flows are developed based on two efficient axisymmetric LB models available in the literature. In accord with their respective original models, the proposed axisymmetric models evolve within the framework of the standard LB method and the source terms contain no gradient calculations. Moreover, the incompressibility conditions are realized with the Hermite expansion, thus the compressibility errors arising in the existing models are expected to be reduced by the proposed incompressible models. In addition, an extra relaxation parameter is added to the Bhatnagar-Gross-Krook collision operator to suppress the effect of the ghost variable and thus the numerical stability of the present models is significantly improved. Theoretical analyses, based on the Chapman-Enskog expansion and the equivalent moment system, are performed to derive the macroscopic equations from the LB models and the resulting truncation terms (i.e., the compressibility errors) are investigated. In addition, numerical validations are carried out based on four well-acknowledged benchmark tests and the accuracy and applicability of the proposed incompressible axisymmetric LB models are verified.

A finite difference solver for incompressible Navier–Stokes flows in complex domains

Modern CFD applications require the treatment of general complex domains to accurately model the emerging flow patterns. In the present work, a new low order finite difference scheme is employed and tested for the numerical solution of the incompressible Navier–Stokes equations in a complex domain described in curvilinear coordinates. A staggered grid discretization is used on both the physical and computational domains. A subgrid based computation of the Jacobian and the metric coefficients of the transformation is used. The incompressibility condition, properly transformed in curvilinear coordinates, is enforced by an iterative procedure employing either a modified local pressure correction technique or the globally defined numerical solution of a general elliptic BVP. Results obtained by the proposed overall solution technique, exhibit very good agreement with other experimental and numerical calculations for a variety of domains and grid configurations. The overall numerical solver effectively treats the general complex domains.

Surface-only liquids

We propose a novel surface-only technique for simulating incompressible, inviscid and uniform-density liquids with surface tension in three dimensions. The liquid surface is captured by a triangle mesh on which a Lagrangian velocity field is stored. Because advection of the velocity field may violate the incompressibility condition, we devise an orthogonal projection technique to remove the divergence while requiring the evaluation of only two boundary integrals. The forces of surface tension, gravity, and solid contact are all treated by a boundary element solve, allowing us to perform detailed simulations of a wide range of liquid phenomena, including waterbells, droplet and jet collisions, fluid chains, and crown splashes.

Localized forms of the LBB condition and a posteriori estimates for incompressible media problems

The inf–sup (or LBB) condition plays a crucial role in analysis of viscous flow problems and other problems related to incompressible media. In this paper, we deduce localized forms of this condition that contain a collection of local constants associated with subdomains instead of one global constant for the whole domain. Localized forms of the LBB inequality imply estimates of the distance to the set of divergence free fields. We use them and deduce fully computable bounds of the distance between approximate and exact solutions of boundary value problems arising in the theory of viscous incompressible fluids. The estimates are valid for approximations, which satisfy the incompressibility condition only in a very weak (integral) form. Another important question considered in the paper is how to select proper measures that should be used in error analysis. We show that such a measure is dictated by the respective error identity and discuss properties of the measure for the Stokes, Oseen, and Navier–Stokes problems.

A turbulence characteristic length scale for compressible flows

ABSTRACTThe current RANS models are generally established and calibrated under incompressible condition and these kinds of models could succeed in predicting many features of incompressible flows. However, these models extended to the high-speed, compressible flows are always less accurate. In the paper, a compressible von Kármán length scale is proposed for compressible flows considering the variable densities. It contains no empirical coefficients and is based on phenomenological theory. In the turbulent kinetic equation, the extra unclosed terms induced by non-constant densities are treated as dissipation terms and the equation is closed algebraically via the introduction of the von Kármán length scale. The original and the proposed von Kármán length scale lead to two different kinds of SAS (scale adaption simulation) models, KDO (turbulence kinetic energy dependent only) and CKDO (compressible KDO), respectively. Compressible mixing layer with significant compressibility is studied within standard k–ϵ, k–ω, KDO turbulence models and their compressible versions. The compressibility effects such as the reduced mixing layer thickness, growth rate and turbulence intensity can be reproduced by CKDO. The new length scale can improve the performances of the model in predicting the mixing layer thickness, stream-wise velocity and Reynolds shear stresses when the convective Mach number is 0.8. Besides, the new length scale also leads to accurate computed growth rate when the convective Mach number ranges from 0.1 to 1.0.

Field emission microplasma actuation for microchannel flows

Microplasmas offer attractive flow control methodology for gas transport in microsystems where large viscous losses make conventional pumping methods highly inefficient. We study microscale flow actuation by dielectric-barrier discharge (DBD) with field emission (FE) of electrons, which allows lowering the operational voltage from kV to a few hundred volts and below. A feasibility study of FE-DBD for flow actuation is performed using 2D particle-in-cell method with Monte Carlo collisions (PIC/MCC) at 10 MHz in nitrogen at atmospheric pressure. The free diffusion dominated, high velocity field emission electrons create a large positive space charge and a body force on the order of 106 N m−3. The body force and Joule heat decrease with increase in dielectric thickness and electrode thickness. The body force also decreases at lower pressures. The plasma body force distribution along with the Joule heating is then used in the Navier–Stokes simulations to quantify the flow actuation in a microchannel. Theoretical analysis and simulations for plasma actuated planar Poiseuille flow show that the gain in flow rate is inversely proportional to Reynolds number. This theoretical analysis is in good agreement with the simulations for a microchannel with closely placed actuators under incompressible conditions. Flow rate of FE-DBD driven 2D microchannel is around 100 ml min−1 mm−1 for an input power of 64 μW mm−1. The gas temperature rises by 1500 K due to the Joule heating, indicating FE-DBD’s potential for microcombustion, micropropulsion and chemical sensing in addition to microscale pumping and mixing applications.

Thermal creep transition stresses and strain rates in a circular disc with shaft having variable density

PurposeThe purpose of this paper is to present study of thermal creep stresses and strain rates in a circular disc with shaft having variable density by using Seth’s transition theory.Design/methodology/approachSeth’s transition theory is applied to the problem of thermal creep transition stresses and strain rates in a thin rotating disc with shaft having variable density by finite deformation. Neither the yield criterion nor the associated flow rule is assumed here. The results obtained here are applicable to compressible materials. If the additional condition of incompressibility is imposed, then the expression for stresses corresponds to those arising from Tresca yield condition.FindingsThermal effect increased value of radial stress at the internal surface of the rotating disc made of incompressible material as compared to tangential stress and this value of radial stress further much increases with the increase in angular speed as compared to without thermal effect. Strain rates have maximum values at the internal surface for compressible material.Originality/valueThe model proposed in this paper is used in mechanical and electronic devices. They have extensive practical engineering application such as in steam and gas turbines, turbo generators, flywheel of internal combustion engines, turbojet engines, reciprocating engines, centrifugal compressors and brake disks.

Internal wave pressure, velocity, and energy flux from density perturbations

Determination of energy transport is crucial for understanding the energy budget and fluid circulation in density varying fluids such as the ocean and the atmosphere. However, it is rarely possible to determine the energy flux field $\mathbf{J} = p \mathbf{u}$, which requires simultaneous measurements of the pressure and velocity perturbation fields, $p$ and $\mathbf{u}$. We present a method for obtaining the instantaneous $\mathbf{J}(x,z,t)$ from density perturbations alone: a Green's function-based calculation yields $p$, and $\mathbf{u}$ is obtained by integrating the continuity equation and the incompressibility condition. We validate our method with results from Navier-Stokes simulations: the Green's function method is applied to the density perturbation field from the simulations, and the result for $\mathbf{J}$ is found to agree typically to within $1\%$ with $\mathbf{J}$ computed directly using $p$ and $ \mathbf{u}$ from the Navier-Stokes simulation. We also apply the Green's function method to density perturbation data from laboratory schlieren measurements of internal waves in a stratified fluid, and the result for $\mathbf{J}$ agrees to within $6\%$ with results from Navier-Stokes simulations. Our method for determining the instantaneous velocity, pressure, and energy flux fields applies to any system described by a linear approximation of the density perturbation field, e.g., to small amplitude lee waves and propagating vertical modes. The method can be applied using our Matlab graphical user interface EnergyFlux.

An improved parallel SPH approach to solve 3D transient generalized Newtonian free surface flows

In this paper, a corrected parallel smoothed particle hydrodynamics (C-SPH) method is proposed to simulate the 3D generalized Newtonian free surface flows with low Reynolds number, especially the 3D viscous jets buckling problems are investigated. The proposed C-SPH method is achieved by coupling an improved SPH method based on the incompressible condition with the traditional SPH (TSPH), that is, the improved SPH with diffusive term and first-order Kernel gradient correction scheme is used in the interior of the fluid domain, and the TSPH is used near the free surface. Thus the C-SPH method possesses the advantages of two methods. Meanwhile, an effective and convenient boundary treatment is presented to deal with 3D multiple-boundary problem, and the MPI parallelization technique with a dynamic cells neighbor particle searching method is considered to improve the computational efficiency. The validity and the merits of the C-SPH are first verified by solving several benchmarks and compared with other results. Then the viscous jet folding/coiling based on the Cross model is simulated by the C-SPH method and compared with other experimental or numerical results. Specially, the influences of macroscopic parameters on the flow are discussed. All the numerical results agree well with available data, and show that the C-SPH method has higher accuracy and better stability for solving 3D moving free surface flows over other particle methods.

Towards direct numerical simulations of low-Mach number turbulent reacting and two-phase flows using immersed boundaries

Most reacting and two-phase flows of practical interest are turbulent but take place at low Mach numbers or under incompressible conditions. In order to investigate the properties of such complex flows with high accuracy but acceptable computing times, a suitable tool for Direct Numerical Simulations (DNS), called DINOSOARS, has been developed. The present article describes the numerical components and methods implemented in this code, together with a detailed verification and validation phase, and finishes with two examples of full-scale simulations. We hope it might be useful as a “verification and validation guideline” for other researchers working on DNS of reacting flows. Since applications of growing complexity are considered by DNS, a Direct Boundary Immersed Boundary Method (DB-IBM) has been implemented, allowing a description of arbitrary geometries on a fixed, but possibly refined, Cartesian mesh. A direct force IBM is implemented as well in DINOSOARS in order to resolve large moving spherical particles (much larger than the Kolmogorov scale) on the grid. Particles below the Kolmogorov scale are treated as point particles, taking into account additionally heat and mass transfer with the continuous flow. The efficient parallelization of the code relies on the open-source library 2DECOMP&FFT. The underlying Poisson equation is solved in a fast and accurate manner by FFT, even for non-periodic boundary conditions. The flexibility of DINOSOARS makes it a very promising tool for analyzing a variety of problems and applications involving turbulent reacting and/or two-phase flows.

Investigation of Pump Impeller Performance using CFD

Objectives: Impeller is the rotating component of a centrifugal pump which imparts kinetic energy on the working fluid. The present work is concentrated on the optimization of geometry of the impeller to increase the efficiency of centrifugal pump. Methods/Statistical Analysis: The L9 orthogonal array table is formed by taking the parameters such as impeller width, number of blades and inlet diameter and outlet blade angle as control factors and efficiency as response. The impeller is modelled using the software Pro_E and the analysed using CFD software Fluent 6.3 by assuming the steady, incompressible and viscous flow condition for the working fluid. Findings: The optimization technique is carried out based on taguchi concept with the aid of CFD software. From the contribution ratio values it is found that the parameter Number of Blades contributes a significant portion in affecting the pump efficiency. Based on the values of pressure, discharge and torque obtained from the software, the corresponding values of efficiencies are calculated. Application/Improvement: Using larger the better quality characteristics, the geometry of the impeller is optimized for maximum efficiencyKeywords: CFD and Optimization, Efficiency, Impeller, Vane