A three-dimensional passage flow analysis method aimed at centrifugal impellers

A partially parabolic procedure is developed to analyze three-dimensional viscous flows through curved ducts of arbitrary cross-section. The procedure, eventually aimed at centrifugal impeller analysis, incorporates a finite-volume method using a strong conservation form of the parabolized Navier-Stokes equations written in arbitrary curvilinear coordinates. Cartesian velocity components and pressure are used as dependent variables. A solution is achieved through pressure corrections which influence velocity semi-implicity. The basic physical elements associated with centrifugal impellers are considered. Laminar flow through 90° bent square duct, turbulent flow in low-aspect-ratio diffusers and subsonic compressible flow through an accelerating rectangular elbow are calculated. Turbulence is accounted for using the k − ε turbulence model. Good correlation between the predictions and experimental data was achieved.     

Computation of three-dimensional parabolic laminar flows

A simple yet accurate method for computing three-dimensional parabolic laminar flow of an incompressible, viscous fluid in a constant area duct is presented. The boundary-layer type nonlinear equations of motion are solved numerically using finite differences and the variable mesh technique. Results of the present calculations are compared with experimental and previous theoretical data for flow through square and rectangular ducts; the agreement is satisfactory.     

Study of incompressible flow separation using primitive variables

The flow in a channel with an asymmetric constriction has been analyzed, using the primitive variables. The objective of this study is to develop a reduced form of the Navier-Stokes equations that is regular at the separation point. The resulting formulation is a semi-elliptic model comprised of the parabolized momentum equations together with an elliptic equation for the pressure distribution. Analytic coordinate transformations are used to provide surface-oriented coordinates with proper resolution for the viscous regions near the channel walls, as well as for the separated-flow regions. Results are obtained for several flow configurations using a semi-implicit numerical method of solution. These are presented in terms of streamling contours and shear-stress and pressure distributions at the channel walls. Comparison with the corresponding Navier-Stokes results shows that the semi-elliptic model performs very well as regards the separation singularity. Some of the qualitative flow features predicted by the available asymptotic analysis for constricted-channel flows are also well reproduced in some of the present results obtained for $\text{Re=1000}$.     

A numerical method for the vorticity-velocity Navier-Stokes equations in two and three dimensions

This paper provides a numerical method for solving the steady-state vorticity-velocity Navier-Stokes equations in two and three dimensions. The vorticity transport equation is considered together with a Poisson equation for the velocity vector, the latter equation being parabolized in time according to the false transient approach. The two vector equations are discretized in time using the implicit Euler time stepping and the delta form of Beam and Warming. A staggered-grid spatial discretization is employed in conjunction with a deferred correction procedure. Second-order-accurate central differences are used to approximate the steady-state residuals, written in conservative form for accuracy reasons, whereas upwind differences are used for the advection terms in the implicit operator, to obtain diagonally-dominant tridiagonal systems. The discrete equations are solved sequentially by means of a robust alternating direction line-Gauss-Seidel iteration procedure combined with a simple multigrid strategy. For the model driven-cavity-flow problem in two and three dimensions, the method is found to be efficient and very accurate. For the first time, the three-dimensional discrete vorticity and velocity fields, computed using a Poisson equation for the velocity vector, are both solenoidal and satisfy their mutual relationship, exactly.     

An implicit marching method for the two-dimensional reduced Navier-Stokes equations at arbitrary mach number

An implicit marching procedure is developed from a parabolized form of the two-dimensional Navier-Stokes equations with allowance for chemical reaction in an effort to predict the flow in highly viscous chemical laser nozzles. The equations governing the fluid dynamics are linearized and solved as a coupled implicit system in a manner which requires no iteration and in which the pressure gradient for internal flows emerges directly as part of the solution without iteration. Accuracy and stability of the method are tested by computing the flow in the entrance region between parallel flat plates and comparing the results to analytical results as well as experimental data. Upon completion of these tests the procedure is used to predict the highly viscous chemically reacting flow in a laser nozzle and inlet duct.     

A Numerical Scheme for a Viscous Shallow Water Model with Friction

We consider a particular viscous shallow water model with topography and friction laws, formally derived by asymptotic expansion from the three-dimensional free surface Navier-Stokes equations. Emphasize is put on the numerical study: the viscous system is regarded as an hyperbolic system with source terms and discretized using a second order finite volume method. New steady states solutions for open channel flows are introduced for the whole model with viscous and friction terms. The proposed numerical scheme is validated against these new benchmarks.     

Reduced order modeling based on POD of a parabolized Navier–Stokes equations model II: Trust region POD 4D VAR data assimilation

A reduced order model based on Proper Orthogonal Decomposition (POD) 4D VAR (Four-dimensional Variational) data assimilation for the parabolized Navier–Stokes (PNS) equations is derived. Various approaches of POD implementation of the reduced order inverse problem are studied and compared including an ad-hoc POD adaptivity along with a trust region POD adaptivity. The numerical results obtained show that the trust region POD 4D VAR provides the best results amongst all the POD adaptive methods tested in all error metrics for the reduced order inverse problem of the PNS equations.     

Calculation of internal viscous flows in axisymmetric ducts at moderate to high reynolds numbers

A formal procedure is presented for parabolizing the Navier-Stokes equations for two dimensional internal flows in artitrary ducts with streamline curvature and streamline divergence. This procedure consists of constructing an orthogonal coordinate system from the potential flow solution and parabolizing the corresponding Navier-Stokes equations using the thin channel approximation. Theoretical arguments based on the fundamental existence theorem and numerical examples are used to demonstrate the validity of this procedure. A new method using conformal mapping based on the Schwartz-Christoffel transformation is presented which obtains the solution of the inverse potential flow problem in a direct manner. A numerical solution algorithm based on the two point box scheme is presented for solving the viscous flow equations. This method is shown to be accurate and stable for flows at moderate to high Reynolds numbers over a wide range of conditions.     

The simulation of 3D unsteady incompressible flows with moving boundaries on unstructured meshes

The development of a computational model for the simulation of three-dimensional unsteady incompressible viscous fluid flows with moving boundaries is presented. The numerical model is based upon the solution of the Navier–Stokes equations on unstructured meshes using the artificial compressibility approach. An ALE formulation is adopted and the equations are discretized using a cell vertex finite volume method. The formulation ensures the satisfaction of the geometric conservation law when the mesh is allowed to move. An implicit time discretization is adopted and a dual time approach is employed. Explicit relaxation is used for the sub-iterations, with multigrid acceleration. For moving geometries, the mesh is deformed by adopting a spring analogy, combined with a wall distance function approach. The numerical procedure is validated on a standard problem and is then used for the simulation of flow over a flexible fish-like body.     

Simulation of isotropic turbulence degeneration based on the large-eddy method

In this paper, the simulation of isotropic turbulence degeneration is studied. The turbulent process is modeled based on filtered three-dimensional unsteady Navier-Stokes equations. For the closure of the main equations, a viscous model of turbulence is used. The problem is solved numerically, i.e., in solving the equation of motion the modified method of fractional steps using compact schemes is employed and the equation for pressure is solved by the Fourier method in combination with matrix factorization. Temporal variations in the kinetic energy of turbulence and changes in the micro scale of turbulence and longitudinal-transverse correlation functions have been obtained. Longitudinal and transverse one-dimensional spectra have been found.     

Chebyshev pseudospectral solution of the incompressible Navier-Stokes equations in curvilinear domains

A pseudospectral method for the calculation of 2-D flows of a viscous incompressible fluid in curvilinear domains is presented. The incompressible Navier-Stokes equations, expressed in terms of the primitive variables velocity and pressure, are solved in a non-orthogonal coordinate system. All the variables are expanded in double truncated series of Chebyshev polynomials. Time integration is performed by an implicit finite differences scheme for both the advective and diffusive terms. The pressure is calculated by the use of a truncated influence matrix involving all the collocation points in the field. A preconditioned iterative method is used to solve the system of linear equations resulting from the pseudospectral Chebyshev approximation. The algorithm is applied to the classical problem of the Green-Taylor vortices in order to check its accuracy; then 2 examples of viscous flow calculation are given in the case of a driven polar cavity and of a 2-D channel.     

Finite Element Simulation of Three-Dimensional Free-Surface Flow Problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet.     

Simulating cyclic artery compression using a 3D unsteady model with fluid–structure interactions

High pulsating blood pressure and severe stenosis make fluid–structure interaction (FSI) an important role in simulating blood flow in stenotic arteries. A three-dimensional nonlinear model with FSI and a numerical method using GFD are introduced to study unsteady viscous flow in stenotic tubes with cyclic wall collapse simulating blood flow in stenotic carotid arteries. The Navier–Stokes equations are used as the governing equations for the fluid. A thin-shell model is used for the tube wall. Interaction between fluid and tube wall is treated by an incremental boundary iteration method. Elastic properties of the tube wall are determined experimentally using a polyvinyl alcohol hydrogel artery stenosis model. Cyclic tube compression and collapse, negative pressure and high shear stress at the throat of the stenosis, flow recirculation and low shear stress just distal to the stenoses were observed under physiological conditions. These critical flow and mechanical conditions may be related to platelet aggregation, thrombus formation, excessive artery fatigue and possible plaque cap rupture. Computational and experimental results are compared and reasonable agreement is found.     

Analytical Investigation of the Dynamics of a Hydraulic Fracture Using the Principle of Incomplete Coupling

An investigation of a self-similar solution of a coupled problem on the creeping flows of a viscous fluid in a hydraulic fracture and the strain and flow in the external poroelastic medium induced by them. The process is governed by injection fluid into a well. The flow in the fracture is described by the Stokes hydrodynamic equations in the approximation of the lubricant layer. The external problem is described by the equations of poroelasticity. The variant of a uniform pressure in the fracture is considered for three-dimensional and two-dimensional cases. In the second case, a self-similar solution can be obtained in an analytical form.     

Shape design optimization of an expansion step in a channel with moving boundaries for a viscous flow

A shape design optimization problem for viscous flows has been investigated in the present study. An analytical shape design sensitivity expression has been derived for a general integral functional by using the adjoint variable method and the material derivative concept of optimization. A channel flow problem with a backward facing step and adversely moving boundary wall is taken as an example. The shape profile of the expansion step, represented by a fourth-degree polynomial, is optimized in order to minimize the total viscous dissipation in the flow field. Numerical discretizations of the primary (flow) and adjoint problems are achieved by using the Galerkin FEM method. A balancing upwinding technique is also used in the equations. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. It is concluded that the proposed general method of solution for shape design optimization problems is applicable to physical systems described by nonlinear equations.     

Application of a posteriori error estimation to finite element simulation of incompressible Navier-Stokes flow

The main goal of this paper is to study adaptive mesh techniques, using a posteriori error estimates, for the finite element solution of the Navier-Stokes equations modeling steady and unsteady flows of an incompressible viscous fluid. Among existing operator splitting techniques, the θ-scheme is used for time integration of the Navier-Stokes equations. Then, a posteriori error estimates, based on the solution of a local system for each triangular element, are presented in the framework of the generalized incompressible Stokes problem, followed by its practical application to the case of incompressible Navier-Stokes problem. Hierarchical mesh adaptive techniques are developed in response to the a posteriori error estimation. Numerical simulations of viscous flows associated with selected geometries are performed and discussed to demonstrate the accuracy and efficiency of our methodology.     

Global PNS solutions for subsonic strong interaction flow over a cone-cylinder-boattail configuration

The solution of the semi-elliptic or so-called parabolized Navier-Stokes equations is considered for large Reynolds numbers and subsonic flows with strong pressure interaction. Flow past a cone-cylinder-boattail configuration is investigated as a prototype of strong viscous-inviscid interaction. A global boundary-layer relaxation procedure is utilized for the formulation of the discrete boundary-value problem. The resulting marching procedure does not require a sub-layer type of approximation. Furthermore, the method does not restrict the step size in the marching direction and is free from any departure effects. Solutions with large recirculation regions are calculated.     

An immersed boundary method based on the lattice Boltzmann approach in three dimensions,with application

The immersed boundary (IB) method originated by Peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid. The Navier–Stokes (N–S) equations in the IB method are usually solved using numerical methods such as FFT and projection methods. Here in our work, the N–S equations are solved by an alternative approach, the lattice Boltzmann method (LBM). Compared to many conventional N–S solvers, the LBM can be easier to implement and more convenient for modeling additional physics in a problem. This alternative approach adds extra versatility to the immersed boundary method. In this paper we discuss the use of a 3D lattice Boltzmann model (D3Q19) within the IB method. We use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle line in a 3D channel and determine a drag scaling law for the sheet. Our main conclusions are: (1) the hybrid method is convergent with first-order accuracy which is consistent with the immersed boundary method in general; (2) the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow.     

Three-dimensional and analytical modeling of microfluidic particle transport in magnetic fluids

We present an analytical model that can predict the three-dimensional (3D) transport of non-magnetic particles in magnetic fluids inside a microfluidic channel coupled with permanent magnets. The magnets produce a spatially non-uniform magnetic field that gives rise to a magnetic buoyancy force on the particles. Resulting 3D trajectories of the particles are obtained by (1) calculating the 3D magnetic buoyancy force exerted on the particles via an analytical distribution of magnetic fields as well as their gradients, together with a nonlinear magnetization model of the magnetic fluids, (2) deriving the 3D hydrodynamic viscous drag force on the particles with an analytical velocity profile of a low Reynolds number ferrohydrodynamic flow in the channel including “wall effect” and magnetoviscous effect of the magnetic fluids, and (3) constituting and solving the governing equations of motion for the particles using the analytical expressions of magnetic buoyancy force and hydrodynamic viscous drag force. We use such a model to study the particles’ trajectories in the channel and investigate the magnitude of their deflections at different flow rates, with different properties of magnetic fluids and different geometrical parameters of the system.     

Coupling of a nonlinear finite element structural method with a Navier-Stokes solver

A new three-dimensional viscous aeroelastic solver is developed in the present work. A well validated full Navier-Stokes code is coupled with a nonlinear finite element plate model. Implicit coupling between the computational fluid dynamics and structural solvers is achieved using a subiteration approach. Computations of several benchmark static and dynamic plate problems are used to validate the finite element portion of the code. This coupled aeroelastic scheme is then applied to the problem of three-dimensional panel flutter. Inviscid and viscous supersonic results match previous computations using the same aerodynamic method coupled with a finite difference structural solver. For the case of subsonic flow, multiple solutions consisting of static, upward and downward deflections of the panel are discussed. The particular solution obtained is shown to be sensitive to the cavity pressure specified underneath the panel.