Solution Of Viscous Flow Research Articles

Inviscid Nonlinear Modeling of Vibration-Induced Acoustic Resonance of a Linear Cascade

A nonlinear model is developed to investigate the blade vibration-induced acoustic resonance in a linear cascade. The Euler equations are solved by using high-order computational aeroacoustics techniques. To satisfy no-penetration wall boundary conditions for vibrating blades, a semi-implicit time-accurate body force model is developed. Leading-edge singularity at resonance is remarkably strong, and so the balance of globally high-order quality (less dissipation) and numerical stability is significant. The adaptive discontinuity-capturing filtering is introduced as a local operation to achieve the good balance. To validate our model, the benchmark case of the Parker acoustic resonance induced by the vibrating cascade of flat plates is revisited. Compared to the linearized theory and viscous flow solution, our inviscid model economically achieves a good prediction both for the unsteady loading and the near-field sound pressure for the small blade vibration amplitude of . When it comes to a larger vibration amplitude of , nonlinearity on the unsteady lift and sound pressure appears in our results, and it cannot be considered in the linearized theory. The results, to some extent, are also consistent with the viscous flow solution. The study well verifies the capability of our nonlinear model in predicting this category of linear and nonlinear sound-vortex/structure interaction problems.

SPH-ALE Scheme for Weakly Compressible Viscous Flow with a Posteriori Stabilization

A highly accurate SPH method with a new stabilization paradigm has been introduced by the authors in a recent paper aimed to solve Euler equations for ideal gases. We present here the extension of the method to viscous incompressible flow. Incompressibility is tackled assuming a weakly compressible approach. The method adopts the SPH-ALE framework and improves accuracy by taking high-order variable reconstruction of the Riemann states at the midpoints between interacting particles. The moving least squares technique is used to estimate the derivatives required for the Taylor approximations for convective fluxes, and also provides the derivatives needed to discretize the viscous flux terms. Stability is preserved by implementing the a posteriori Multi-dimensional Optimal Order Detection (MOOD) method procedure thus avoiding the utilization of any slope/flux limiter or artificial viscosity. The capabilities of the method are illustrated by solving one- and two-dimensional Riemann problems and benchmark cases. The proposed methodology shows improvements in accuracy in the Riemann problems and does not require any parameter calibration. In addition, the method is extended to the solution of viscous flow and results are validated with the analytical Taylor–Green, Couette and Poiseuille flows, and lid-driven cavity test cases.

Gas Flow in Microchannels and Nanochannels With Variable Cross Section for All Knudsen and All Mach Number Values

AbstractThe analytical solution for steady viscous pressure-driven compressible isothermal gas flow through micro- and nanochannels with variable cross section for all Knudsen and all Mach number values is presented in this paper. The continuum one-dimensional governing equations are solved using the friction factor that is established in a special way to provide solutions for mass flow rate, pressure, and velocity distribution through the microchannels and nanochannels in the entire rarefaction regime. The friction factor, defined by the general boundary condition and generalized diffusion coefficient proposed by Beskok and Karniadakis (1999, “A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales,” J. Microscale Thermophys. Eng., 3, pp. 43–77), spreads the solution application to all rarefaction regimes from continuum to free molecular flow. The correlation between the product of friction factor and Reynolds number (Poiseuille number) and Knudsen number is established explicitly in the paper. Moreover, the obtained solution includes the inertia effect, which allows the application of the solution to both subsonic and supersonic gas flows, which was not shown earlier. The presented solution confirms the existence of the Knudsen minimum in the diverging, converging, and microchannels and nanochannels with constant cross section. The proposed solution is verified by comparison with experimental, analytical, and numerical results available in literature.

Analytical and Numerical Solution of Viscous Fluid Flow with the Effects of Thermal Radiation and Chemical Reaction Past a Vertical Porous Surface

Magnetohydrodynamic (MHD) unsteady flow of incompressible, electrically conducting, and viscous fluid passing through a vertical porous plate is explored in this investigation where the impact of thermal radiation and chemical reaction are presented. A nonlinear two-dimensional unsteady flow model is created to decompose heat and mass transfer of this fluid, which is shaped by taking appropriate boundary conditions. The governing equations are solved analytically with satisfactory boundary conditions by using a two-term perturbation technique. The analytical outcomes are illustrated graphically by using MATHEMATICA 7.0. The fundamental equations are also solved numerically by utilizing explicit finite difference methodology (EFDM) with the help of Compaq Visual Fortran 6.6a. The numerical arrangement is also depicted graphically to examine the outcomes. The acquired pertinent parameters are analysed conveniently, including thermal Grashof number, thermal radiation parameter, and chemical reaction parameter, the permeability of the porous medium, magnetic parameter, Prandtl number, modified Grashof number, and Schmidt number. It is observed from our experiment that the velocity and concentric profiles decline when the reaction rate between the molecules is high. Furthermore, a suitable validation has been done with the previously published articles, and a favorable agreement is observed.

A velocity-vorticity method for highly viscous 3D flows with application to digital rock physics

In this article, we present a numerical iterative method for the solution of internal viscous and incompressible flows in real porous three-dimensional bodies at their pore scale. We use the penalized formulation of the problem involving velocity and vorticity: an operator splitting allows to split apart the diffusion (inherited from Stokes equation) and the penalization phenomena (which takes into account the solid matrix). By means of the numerical analysis of the splitting, we exhibit the penalization coefficient which is actually effective. This method allows to deal only with fast-evaluation operators, that is to say scaling at most as O(nlog⁡n) where n is the number of underlying grid points, such as straightforward computations of finite differences schemes or FFT solver. The numerical analysis and implementation solutions are presented, and validated on various digital rock physics geometries acquired by micro-tomography, using numerical and physical diagnostics. To enforce this validation, we also present permeability estimations of several porous samples. The simulation of transport of passive and active scalars is finally investigated in order to perform the practical upscaling to 1D models of transport and diffusion at the Darcy scale.

Geometry definition and performance assessment of Tesla turbines for ORC

The Tesla turbine - also known as friction, viscous or bladeless turbine - is a peculiar expander, which generates power through viscous entrainment. In the last years, it has gained a renewed appeal due to the rising of distributed power generation applications. Indeed, this expander is not suitable to large size power generation, but it could become a breakthrough technology in the low power ranges, due to its characteristics of low cost and reliability. The current study presents a design approach to the Tesla turbine, applied to organic working fluids (R1233zd(E), R245fa, R1234yf, n-Hexane). Three fundamental geometric parameters are identified (rotor channel width/inlet diameter ratio, rotor outlet/inlet diameter ratio, throat width ratio) and their effects on the performance are analysed. The geometry of the turbine has been defined and the assessment of the performance potential is run, applying a 2D code for the viscous flow solution, considering real compressible fluid properties. For all the investigated working fluids, an efficiency higher than 60% has been achieved, with the defined geometry, under suitable thermo fluid-dynamic conditions.

The singular hydrodynamic interactions between two spheres in Stokes flow

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centers, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs), which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by the use of a bipolar coordinate system, we derive the stream function for the flow as the Reynolds number (Re) tends to zero and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behavior of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon the known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with the existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix R in a number of situations in which positivity is destroyed for multipole/perturbative methods.

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

As an extension of a fourth-order compact gas kinetic scheme (GKS) on structured mesh [24], this work is about the development of a third-order compact GKS on unstructured mesh for the compressible Euler and Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic evolution solution, the time-dependent gas distribution function at a cell interface in GKS provides not only the flux function and its time derivative, but also the time accurate flow variables there at the next time level. As a result, besides updating the conservative flow variables inside each control volume through the interface fluxes, the cell averaged first-order spatial derivatives of flow variables can be obtained as well using the updated flow variables at the closed cell interfaces around that cell through the divergence theorem. Therefore, with the cell-averaged flow variables and their first-order spatial derivatives inside each cell, the Hermite WENO (HWENO) techniques can be naturally implemented for the compact high-order reconstruction at the beginning of the next time step. Following the reconstruction technique in [64], a new HWENO reconstruction on triangular mesh is designed in the current scheme. Combined with the two-stage temporal discretization and second-order time accurate flux function, a third-order compact scheme on unstructured mesh has been constructed. Accurate solutions can be obtained for both inviscid and viscous flows without sensitive dependence on the quality of triangular mesh. The robustness and accuracy of the scheme have been validated through many cases, including strong shocks in the hypersonic viscous flow and smooth Navier-Stokes solution.

Application Progress of Computational Fluid Dynamic Techniques for Complex Viscous Flows in Ship and Ocean Engineering

Complex flow around floating structures is a highly nonlinear problem, and it is a typical feature in ship and ocean engineering. Traditional experimental methods and potential flow theory have limitations in predicting complex viscous flows. With the improvement of high-performance computing and the development of numerical techniques, computational fluid dynamics (CFD) has become increasingly powerful in predicting the complex viscous flow around floating structures. This paper reviews the recent progress in CFD techniques for numerical solutions of typical complex viscous flows in ship and ocean engineering. Applications to free-surface flows, breaking bow waves of high-speed ship, ship hull–propeller–rudder interaction, vortex-induced vibration of risers, vortex-induced motions of deep-draft platforms, and floating offshore wind turbines are discussed. Typical techniques, including volume of fluid for sharp interface, dynamic overset grid, detached eddy simulation, and fluid–structure coupling, are reviewed along with their applications. Some novel techniques, such as high-efficiency Cartesian grid method and GPU acceleration technique, are discussed in the last part as the future perspective for further enhancement of accuracy and efficiency for CFD simulations of complex flow in ship and ocean engineering.

A modified Taylor Galerkin scheme for turbulent compressible flow simulation

As already demonstrated by different authors, the Taylor-Galerkin (TG) scheme, in the context of the Finite Element Method (FEM), is particularly suitable for the solution of supersonic flows. The TG scheme, using hexahedral finite elements with analytical evaluation of element matrices, is applied in this work. Tools to avoid locking and a shock capturing technique for the solution of supersonic viscous and non-viscous compressible flows are also employed. However, TG scheme usually presents instabilities in subsonic flows. Even in cases in which the free stream Mach number corresponds to supersonic flows, there are always flow regions, specifically near the walls of the immersed obstacles, where the speed is lower and the local Mach number corresponds to a subsonic flow. The Characteristic Based Split (CBS) scheme was developed in order to obtain a single method to improve the behavior with respect to TG method in subsonic and supersonic regimes. In the last two decades some works have shown advantages in convergence rates of the CBS method when compared to the TG algorithm. However, simulation time increases in the CBS method since split operations, typical of this algorithm, imply in additional element loops. In this paper a hybrid algorithm called Modified-Taylor-Galerkin scheme (MTG) is proposed. This algorithm presents advantages with respect to TG and CBS schemes in terms of convergence properties and computational processing time. In order to get an efficient algorithm, the element matrices are analytically integrated. This is performed with two different approaches. In the first approach the inverse matrix and the determinant of the Jacobian matrix at element level are evaluated with a reduced integration form, using the point located in the center of the element for mass, convective, diffusive and stabilization element matrices; all these matrices are integrated analytically. In the second approach, mass and convective matrices are calculated by a complete integration scheme (including the inverse matrix and the determinant of the Jacobian matrix at element level in the analytical expression to be integrated) and the diffusive and stabilization matrices are calculated with a reduced integration form, using the point located at the center of the element to calculate the inverse matrix and the determinant of the Jacobian matrix at element level. Finally, this work incorporates the Spalart-Allmaras (S-A) turbulence model using a conservative version of the transport equation, as proposed by the authors of the original S-A model in a later paper. Algorithms are tested to determine convergence rate improvements in both laminar and turbulent cases and for different Mach numbers (supersonic, transonic and subsonic flows).&nbsp

Solution of viscous flow in a rectangular region by using the hybrid finite volume scheme

Solution of viscous flow in a rectangular region by using the hybrid finite volume scheme

High-fidelity solver on polyhedral unstructured grids for low-Mach number compressible viscous flow

In this article, we developed an unstructured fluid solver based on finite volume framework for the low-Mach number compressible flows. The present method, so-called FVMS3 (Finite Volume method based on Merged Stencil with 3rd-order reconstruction) formulates two different numerical procedures for spatial reconstructions based on the quadratic polynomial which is performed by using least-square approximations on a merged stencil. In order to improve the reconstruction for discontinuities, we propose the limiting projection approach and smoothness adaptive fitting (SAF) scheme to suppress the numerical oscillation and limit the numerical dissipation. The resulting discretization algorithm that combines FVMS3 with SAF-based limiting projection scheme has third-order accuracy and resolves both smooth and non-smooth solutions with excellent quality. Additionally, a novel numerical model has been proposed by introducing the advection upstream splitting method (AUSM) flux into the pressure projection formulation which results in a unified scheme that works uniformly up to the incompressible limit. The fluid solver that integrates all above new efforts provides high-fidelity solutions for compressible viscous flows particularly for the low Mach regime. The performance of this new solver is verified by numerous benchmark tests. Our numerical results show that the proposed scheme gives accurate and robust solutions for a wide spectrum of test problems.

Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls

• We consider the temporal stability of multiple solutions for flows in a porous channel with moving walls for the first time. • We construct second order finite difference schemes for eigenproblems associated with the linear stability analysis . • We verify the linear stability results for the first time by solving the perturbed nonlinear time dependent equation. In this paper, a viscous, incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is considered. We present multiple symmetric steady-state solutions of this flow problem at several different expanding ratios, and use the linear stability theory to analyse the temporal stability for these solutions under symmetric, antisymmetric and general perturbations. We construct second order finite difference schemes for the eigenvalue problems with boundary conditions associated with those perturbations, and observe that most of these solutions which are stable under symmetric perturbations are unstable under antisymmetric perturbations. Furthermore, we verify the linear stability analysis results by directly solving the original perturbed nonlinear time dependent problem, and find that both stability results are consistent.

Large-Amplitude Intermittent Swimming in Viscous and Inviscid Flows

Recently, it was demonstrated in an inviscid flow that burst-and-coast or intermittent swimming can save energy when compared with continuous swimming and that the energy savings are maximized for large-amplitude pitching motions where flow separation is likely to occur. This paper examines the effects of flow separation on altering inviscid flow predictions. As such, viscous and inviscid flow simulations are presented of a hydrofoil pitching intermittently with large-amplitude motions. It is observed that leading-edge vortex formation and shedding in a viscous flow significantly alter the wake dynamics from the inviscid flow solutions where only trailing-edge shedding is modeled. Moreover, the inviscid flow solutions predict higher peak force production, lower cost of transport, and lower optimal duty cycles than the viscous flow solutions. Despite these differences, the trends in the force production and energetics seen in a viscous flow are well captured by the inviscid flow simulations. Importantly, both predict energy savings on the order of 10–30% for intermittent swimming, and the energy savings increase when the amplitude of motion is increased even when significant leading-edge separation occurs.

Local existence of the generalized solution for three-dimensional compressible viscous flow of micropolar fluid with cylindrical symmetry

In this work, the three-dimensional model for the compressible micropolar fluid flow is considered, whereby it is assumed that the fluid is viscous, perfect, and heat conducting. The flow between two coaxial thermoinsulated cylinders, which leads to a cylindrically symmetric model with homogeneous boundary data for velocity, microrotation, and heat flux, is analyzed.The corresponding PDE system is formulated in the Lagrangian setting, and it is proven that this system has a generalized solution locally in time.

Analytical simulation of magneto-hemodynamic flow in a semi-porous channel using the Polynomial Least Squares Method

The present article proposes a new analytical approximate solution for the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field, solution obtained by using the Polynomial Least Squares Method (PLSM). A comparison of our approximate solutions obtained by PLSM with previously computed approximate solutions illustrates the accuracy of our method. A discussion of the effects of the parameters Re (the Reynolds number) and Ha (the Hartmann number) on the blood flow velocity is included.

Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

SummaryIn this work, a time‐accurate integration of the discontinuous Galerkin space‐discretized Navier‐Stokes equations is performed exploiting the matrix‐free (MF) approach to speed up the solution process of the modified extended backward differentiation formulae (MEBDF) schemes. MEBDF are high‐order accurate implicit multistep schemes composed by three nonlinear stages. The proposed algorithm consists in solving the resulting nonlinear system of each stage with a preconditioned MF Newton/Krylov method using a frozen preconditioner strategy to improve its efficiency. Numerical results for compressible inviscid and viscous test cases, both with a known analytical solution, aim at assessing the performance of the proposed MF‐MEBDF algorithm, by comparing it with the one obtained by using its matrix‐explicit counterpart or the explicit strong stability preserving Runge‐Kutta scheme. In particular, the influence of some relevant physical (low‐speed flows) and discretization (aspect ratio, polynomial degree) aspects on the performance of the different time integration schemes are investigated, highlighting the pros and cons of the proposed algorithm and its effectiveness in solving nonstiff and stiff systems. Furthermore, the scalability in parallel computations of the proposed algorithm is investigated and, finally, its potential for efficient long‐time simulations is demonstrated computing a laminar vortex shedding behind a circular cylinder at different Reynolds numbers and by comparing its effectiveness with that of the strong stability preserving Runge‐Kutta scheme.

Design optimisation of separate-jet exhausts for the next generation of civil aero-engines

ABSTRACTThe next generation of civil large aero-engines will employ greater bypass ratios compared with contemporary architectures. This results in higher exchange rates between exhaust performance and specific fuel consumption (SFC). Concurrently, the aerodynamic design of the exhaust is expected to play a key role in the success of future turbofans. This paper presents the development of a computational framework for the aerodynamic design of separate-jet exhaust systems for civil aero-engines. A mathematical approach is synthesised based on class-shape transformation (CST) functions for the parametric geometry definition of gas-turbine exhaust components such as annular ducts and nozzles. This geometry formulation is coupled with an automated viscous and compressible flow solution method and a cost-effective design space exploration (DSE) approach. The framework is deployed to optimise the performance of a separate-jet exhaust for very-high-bypass ratio (VHBR) turbofan engine. The optimisations carried out suggest the potential to increase the engine’s net propulsive force compared with a baseline architecture, through optimum exhaust re-design. The proposed method is able to identify and alleviate adverse flow-features that may deteriorate the aerodynamic behaviour of the exhaust system.

Sloshing-Coupled Ship Motion Algorithm for Estimation of Slosh-Induced Pressures

The effect of coupling between sloshing and ship motions in the evaluation of slosh-induced interior pressures is studied. The coupling between sloshing loads and ship motions is modelled through a hybrid algorithm which combines a potential flow solution based on transient Green function for the external ship hydrodynamics with a viscous flow solution based on a multiphase interface capturing volume of fluid (VOF) technique for the interior sloshing motion. The coupled algorithm accounts for full nonlinear slosh forces while the external forces on the hull are determined through a blended scheme of linear radiation-diffraction with nonlinear Froude-Krylov and restoring forces. Consideration of this level of nonlinearities in ship motions is found to have non-negligible effects on the slosh-coupled responses and slosh-induced loads. A scheme is devised to evaluate the statistical measure of the pressures through long-duration simulation studies in extreme irregular waves. It is found that coupling significantly influences the tank interior pressures, and the differences in the pressures between coupled and uncoupled cases can be as much as 100% or more. To determine the RAO over the frequency range needed for the simulation studies in irregular waves, two alternative schemes are proposed, both of which require far less computational time compared to the conventional method of finding RAO at each frequency, and the merits of these are discussed.

Comparison of numerical methods for the solution of viscous incompressible and low Mach number compressible flow

The novelty of the paper is the application of the discontinuous Galerkin method (DGM) to the solution of viscous compressible flow with low Mach number (M≈10−2−10−3). The compressible Navier–Stokes equations are written in the conservative form using conservative variables. They are discretized by the DGM in space. In time we use either the backward difference formula (BDF) or the discontinuous Galerkin method. The resulting technique is called the space-time discontinuous Galerkin method (STDGM).The goal of the presented analysis is to prove the ability of the developed DGM to solve viscous compressible low Mach number flows described by the compressible Navier–Stokes equations without any modification and to obtain results comparable with simulations of incompressible viscous flow described by the incompressible Navier–Stokes equations, solved by the finite element method (FEM). We present numerical results for a flow in a simplified channel of the human vocal tract, for a flow around the airfoil NACA0012 and moreover for flow-induced airfoil vibrations. The results prove the robustness of the discontinuous Galerkin method with respect to the low Mach number.