Three-Dimensional Time-Marching Inviscid and Viscous Solutions for Unsteady Flows Around Vibrating Blades

1994 ◽  
Vol 116 (3) ◽  
pp. 469-476 ◽  
Author(s):  
L. He ◽  
J. D. Denton
Keyword(s):  
Thin Layer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Correction Method ◽  
Navier Stokes ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Blade Passage ◽  
Time Marching ◽  
Viscous Solutions

A three-dimensional nonlinear time-marching method of solving the thin-layer Navier–Stokes equations in a simplified form has been developed for blade flutter calculations. The discretization of the equations is made using the cell-vertex finite volume scheme in space and the four-stage Runge–Kutta scheme in time. Calculations are carried out in a single-blade-passage domain and the phase-shifted periodic condition is implemented by using the shape correction method. The three-dimensional unsteady Euler solution is obtained at conditions of zero viscosity, and is validated against a well-established three-dimensional semi-analytical method. For viscous solutions, the time-step limitation on the explicit temporal discretization scheme is effectively relaxed by using a time-consistent two-grid time-marching technique. A transonic rotor blade passage flow (with tip-leakage) is calculated using the present three-dimensional unsteady viscous solution method. Calculated steady flow results agree well with the corresponding experiment and with other calculations. Calculated unsteady loadings due to oscillations of the rotor blades reveal some notable three-dimensional viscous flow features. The feasibility of solving the simplified thin-layer Navier–Stokes solver for oscillating blade flows at practical conditions is demonstrated.

scholarly journals Three Dimensional Time-Marching Inviscid and Viscous Solutions for Unsteady Flows Around Vibrating Blades

1993 ◽  
Author(s):  
L. He ◽  
J. D. Denton
Keyword(s):  
Thin Layer ◽  
Stokes Equations ◽  
Linear Time ◽  
Correction Method ◽  
Navier Stokes ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Blade Passage ◽  
Time Marching ◽  
Viscous Solutions

A 3-dimensional non-linear time-marching method of solving the thin-layer Navier-Stokes equations in a simplified form has been developed for blade flutter calculations. The discretization of the equations is made using the cell-vertex finite volume scheme in space and the 4-stage Runge-Kutta scheme in time. Calculations are carried out in a single-blade-passage domain and the phase-shifted periodic condition is implemented by using the shape correction method. The 3-D unsteady Euler solution is obtained at conditions of zero viscosity, and is validated against a well-established 3-D semi-analytical method. For viscous solutions, the time-step limitation on the explicit temporal discretization scheme is effectively relaxed by using a time-consistent two-grid time-marching technique. A transonic rotor blade passage flow (with tip-leakage) is calculated using the present 3-D unsteady viscous solution method. Calculated steady flow results agree well with the corresponding experiment and with other calculations. Calculated unsteady loadings due to oscillations of the rotor blades reveal some notable 3-D viscous flow features. The feasibility of solving the simplified thin-layer Navier-Stokes solver for oscillating blade flows at practical conditions is demonstrated.

Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

1969 ◽  
Vol 37 (4) ◽  
pp. 727-750 ◽  
Author(s):  
Gareth P. Williams
Keyword(s):  
Poisson Equation ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Wave Flow ◽  
Trigonometric Interpolation ◽  
Time Step ◽  
Navier Stokes Equations ◽  
The Poisson Equation

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

scholarly journals Three-Dimensional Simulation of Fluid–Structure Interaction Problems Using Monolithic Semi-Implicit Algorithm

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 94 ◽  
Author(s):  
Cornel Marius Murea
Keyword(s):  
Stokes Equations ◽  
Fluid Structure Interaction ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Arbitrary Lagrangian Eulerian ◽  
Time Step ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Dimensional Simulation ◽  
Updated Lagrangian

A monolithic semi-implicit method is presented for three-dimensional simulation of fluid–structure interaction problems. The updated Lagrangian framework is used for the structure modeled by linear elasticity equation and, for the fluid governed by the Navier–Stokes equations, we employ the Arbitrary Lagrangian Eulerian method. We use a global mesh for the fluid–structure domain where the fluid–structure interface is an interior boundary. The continuity of velocity at the interface is automatically satisfied by using globally continuous finite element for the velocity in the fluid–structure mesh. The method is fast because we solve only a linear system at each time step. Three-dimensional numerical tests are presented.

Numerical Solution of Incompressible Unsteady Flows in Turbomachinery

2000 ◽  
Vol 123 (3) ◽  
pp. 680-685 ◽  
Author(s):  
L. He ◽  
K. Sato
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Unsteady Flows ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Flow Solver ◽  
Time Stepping ◽  
Incompressible Viscous Flow ◽  
Dual Time Stepping

A three-dimensional incompressible viscous flow solver of the thin-layer Navier-Stokes equations was developed for the unsteady turbomachinery flow computations. The solution algorithm for the unsteady flows combines the dual time stepping technique with the artificial compressibility approach for solving the incompressible unsteady flow governing equations. For time accurate calculations, subiterations are introduced by marching the equations in the pseudo-time to fully recover the incompressible continuity equation at each real time step, accelerated with a multi-grid technique. Computations of test cases show satisfactory agreements with corresponding theoretical and experimental results, demonstrating the validity and applicability of the present method to unsteady incompressible turbomachinery flows.

Application of a Navier–Stokes Analysis to Flows Through Plane Cascades

1986 ◽  
Vol 108 (1) ◽  
pp. 103-111 ◽  
Author(s):  
O. Scha¨fer ◽  
H.-H. Fru¨hauf ◽  
B. Bauer ◽  
M. Guggolz
Keyword(s):  
Thin Layer ◽  
Stokes Equations ◽  
Density Ratio ◽  
Separated Flows ◽  
Stability And Convergence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fully Implicit ◽  
First Results ◽  
Viscous Solutions

A newly developed method is used to compute a variety of laminar/turbulent, attached/separated flows through plane turbine or compressor cascades. The thin-layer or full Navier–Stokes equations are solved in a 2-D or quasi-2-D/quasi-3-D form taking into account variable axial velocity density ratio/cascade aspect ratio. The turbulence is modeled by the Baldwin–Lomax algebraic two-layer eddy viscosity approach. Improved mesh generation and discretization techniques are introduced. A fully implicit formulation of the flow problem is developed which ensures high stability and convergence. Numerous quantitative comparisons of viscous solutions with experiments and other existing solutions are performed to validate the method. First results on the applicability of the thin-layer assumption are included.

Strongly Coupled Fluid-Structure Interactions via a New Navier-Stokes Method for Prediction of Vortex-Induced Vibrations

2005 ◽  
Author(s):  
Yannis Kallinderis ◽  
Hyung Taek Ahn
Keyword(s):  
Stokes Equations ◽  
Computational Cost ◽  
Three Dimensional ◽  
Turbulence Models ◽  
Design Tool ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Vortex Induced Vibrations ◽  
Offshore Industry

Numerical prediction of vortex-induced vibrations requires employment of the unsteady Navier-Stokes equations. Current Navier-Stokes solvers are quite expensive for three-dimensional flow-structure applications. Acceptance of Computational Fluid Dynamics as a design tool for the offshore industry requires improvements to current CFD methods in order to address the following important issues: (i) stability and computation cost of the numerical simulation process, (ii) restriction on the size of the allowable time-step due to the coupling of the flow and structure solution processes, (iii) excessive number of computational elements for 3-D applications, and (iv) accuracy and computational cost of turbulence models used for high Reynolds number flow. The above four problems are addressed via a new numerical method which employs strong coupling between the flow and the structure solutions. Special coupling is also employed between the Reynolds-averaged Navier-Stokes equations and the Spalart-Allmaras turbulence model. An element-type independent spatial discretization scheme is also presented which can handle general hybrid meshes consisting of hexahedra, prisms, pyramids, and tetrahedral.

Evaluation of numerical diffusion of the finite volume method when modelling surface waves

2019 ◽  
pp. 106-119
Author(s):  
Елена Сергеевна Тятюшкина ◽  
Андрей Сергеевич Козелков ◽  
Андрей Александрович Куркин ◽  
Вадим Викторович Курулин ◽  
Валентин Робертович Ефремов ◽  
...  
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Calculation Method ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Numerical Diffusion ◽  
Volume Method

Обсуждается применение метода конечных объемов при решении уравнений Навье-Стокса для моделирования поверхностных волн. Сформулирована задача о распространении поверхностных волн, которая используется для оценки численной диффузии в решении уравнений Навье-Стокса. Предлагается методика оценки численной диффузии, выражаемой коэффициентом уменьшения амплитуды волны при прохождении ею одной своей длины (коэффициентом затухания). Дана оценка размеров сетки и шага по времени, выраженных в безразмерных величинах относительно параметров волны, необходимых для обеспечения приемлемого значения коэффициента затухания. Показана степень влияния каждого из сеточных параметров на увеличение коэффициента затухания. The application of numerical simulation methods based on the solution of the full three-dimensional Navier-Stokes equations for modelling of wave propagation on the water surface requires the construction of a grid model containing countable nodes throughout the entire volume of water medium. Insufficient grid resolution leads to insufficient detailing of the fields of velocity and pressure, as well as volume fraction of the liquid, which increases the numerical diffusion of the method and, ultimately, leads to an underestimation of oscillation amplitudes of the medium. A large time step also results in a “blurring” of the solution and significantly reduces its stability, especially when using the schemes which compress the front of the media interface. This paper presents the results of an assessment of acceptable grid sizes and time steps expressed in dimensionless parameters with respect to the wave parameters necessary to ensure accuracy of the solution sufficient for geophysical applications. The estimate is given for the method of calculating three-dimensional multiphase flows with a free surface based on solving the Navier-Stokes equations in a one-velocity approximation based on a completely implicit connection between velocity and pressure using the finite volume method. The finite volume method for the numerical solution of the Navier-Stokes equations is implemented for use on arbitrary unstructured grids. The methodology for estimation of numerical diffusion of the calculation method is proposed. This estimation is expressed as a percentage of the wave amplitude decrease at the distance equal to the one wavelength. In turn the methodology is based on the parameters entered to estimate the acceptable grid sizes and time step for the calculation method. Based on the described methodology, the results of the estimation of the grid resolution in the horizontal and vertical directions, the estimation of the time step, and the evaluation of the influence of the discretization scheme of the convective term are presented.

scholarly journals Four free surface algorithms for the 3D Navier–Stokes equations

2015 ◽  
Vol 17 (6) ◽  
pp. 845-856 ◽  
Author(s):  
Nils Reidar B. Olsen
Keyword(s):  
Wave Equation ◽  
Numerical Methods ◽  
Water Surface ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Free Water ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Free Water Surface

Four algorithms are described for computing a steady free water surface with the solution of the three-dimensional (3D) Navier–Stokes equations. The numerical methods are used in hydraulic engineering cases, typically spillways and river modelling. The algorithms were tested against a laboratory experiment of a v-shaped broad-crested weir. The complex geometry of the weir introduced three-dimensional effects, which the numerical methods handled with varying degrees of success. One of the methods tested was the classical volume of fluid (VOF) approach, implemented in the OpenFOAM software with a fixed grid. The other three algorithms used an adaptive grid that followed the free water surface. These methods were coded in the SSIIM 2 program and were based on water continuity, pressure differences and an implicit solution of the diffusive wave equation. The VOF method gave the best results compared with the experiments. However, this method requires a very short time step. Two of the investigated methods compute the water surface location implicitly and can therefore use a much longer time step. The method based on the diffusive wave equation has the disadvantage that the results depend on a calibrated friction factor. All four methods predicted the water depth over the weir with an average accuracy below 14%.

Numerical Study of Vortex Shedding From a Circular Cylinder in Linear Shear Flow

1999 ◽  
Vol 121 (2) ◽  
pp. 460-468 ◽  
Author(s):  
A. Mukhopadhyay ◽  
P. Venugopal ◽  
S. P. Vanka
Keyword(s):  
Circular Cylinder ◽  
Stokes Equations ◽  
Numerical Study ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Slip Condition ◽  
Navier Stokes ◽  
Finite Volume Scheme ◽  
Constant Frequency ◽  
Wall Boundary Conditions

A three-dimensional numerical simulation of linearly sheared flow past a circular cylinder has been performed for a shear parameter β of 0.02 and a mean Reynolds number of 131.5. A cylinder of 24 diameters span is considered. A second-order accurate finite volume scheme is used to integrate the unsteady Navier-Stokes equations. Present computations confirm both qualitatively and quantitatively, the aspects of cellular shedding as reported by several investigators through experimental studies. Up to five constant frequency cells of obliquely shedding vortices are observed. The nondimensional frequencies of these cells are observed to be lower than those given by parallel shedding correlations at the equivalent Reynolds numbers. It is also observed that the cell boundaries continuously move in time. Detailed distributions of vorticity and velocity components are presented to describe the flow. The influence of end-wall boundary conditions is studied by computing two cases, one with free-slip condition, and the other with no-slip condition on disks of radius of five cylinder diameters.

A Fully Three-Dimensional Inverse Method for Turbomachinery Blading With Navier-Stokes Equations

1998 ◽  
Author(s):  
Zhengming Wang ◽  
Ruixian Cai ◽  
Hongji Chen ◽  
Dong Zhang
Keyword(s):  
Inverse Problem ◽  
Stokes Equations ◽  
Interpolation Method ◽  
Three Dimensional ◽  
Curvilinear Coordinates ◽  
Navier Stokes ◽  
Special Treatment ◽  
Blade Profile ◽  
Navier Stokes Equations ◽  
Time Marching

A new numerical method for solving fully three-dimensional inverse shape design problem of turbomachinery blading has been developed. The general inverse problem refers to the problem in which the pressure distributions on suction and pressure surfaces of blade are given, but the corresponding blade profile is unknown. In this paper, the calculations are based on the 3D Navier-Stokes equations expressed in terms of nonorthogonal curvilinear coordinates and corresponding nonorthogonal velocity components, and the explicit time marching algorithm and Baldwin-Lomax turbulence model are adopted. A special treatment for boundary conditions on blade surfaces is employed to satisfy the given pressure distribution. In computational process, an initial blade profile is supposed at starting, and then the blade surfaces will move regularly with time steps in the time marching process until the convergence is reached. The movement velocities at every point of blade surfaces are obtained from the solution of the Navier-Stokes equations. After each revision of the blade profile, the grid is reconstructed, and the aerodynamic parameters need to be transferred between the old and new grid points by an accurate interpolation method. Thus the viscous inverse problem is solved in a new process. The computational results for two test cases indicate that the method presented in this paper is very effective.

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