Optimization Using Arbitrary Lagrangian–Eulerian Formulation of the Navier–Stokes Equations

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
Eysteinn Helgason ◽  
Siniša Krajnović
Keyword(s):  
Stokes Equations ◽  
Optimization Method ◽  
Navier Stokes ◽  
Lagrangian Description ◽  
Computational Domain ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Mesh Motion ◽  
Eulerian Description ◽  
Eulerian Formulation

In this paper, we present a new shape optimization method by using sensitivities obtained from the Arbitrary Lagrangian–Eulerian (ALE) form of the Navier–Stokes equations. In the ALE description, the nodes of the computational domain may be moved with the fluid as in the Lagrangian description, held fixed in space as in the Eulerian description, or moved in some arbitrary way in between. Applying the adjoint method with respect to mesh motion allows the whole sensitivity field for the shape changes to be calculated using only two solver calls, a primal solver call and an adjoint solver call. We show that the sensitivities with respect to the mesh motion can be calculated in a postprocessing step to the primal and adjoint flow simulations. The resulting ALE sensitivities are compared to sensitivities obtained using a finite difference approach. Finally, the sensitivities are coupled to a mesh motion smoothing algorithm, and a duct is optimized with respect to the total pressure drop using the proposed method.

Numerical Simulations of Nonlinear Post-Critical Vibrations of an Airfoil After Flutter Instability

2011 ◽  
Author(s):  
Jaromi´r Hora´cˇek ◽  
Miloslav Feistauer ◽  
Petr Sva´cˇek
Keyword(s):  
Numerical Simulations ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Mass Matrix ◽  
Vertical Direction ◽  
Navier Stokes ◽  
Computational Domain ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Classical Flutter

The contribution deals with the numerical simulation of the flutter of an airfoil with three degrees of freedom (3-DOF) for rotation around an elastic axis, oscillation in the vertical direction and rotation of a flap. The finite element (FE) solution of two-dimensional (2-D) incompressible Navier-Stokes equations is coupled with a system of nonlinear ordinary differential equations describing the airfoil vibrations with large amplitudes taking into account the nonlinear mass matrix. The time-dependent computational domain and a moving grid are treated by the Arbitrary Lagrangian-Eulerian (ALE) method and a suitable stabilization of the FE discretization is applied. The developed method was successfully tested by the classical flutter computation of the critical flutter velocity using NASTRAN program considering the linear model of vibrations and the double-lattice aerodynamic theory. The method was applied to the numerical simulations of the post flutter regime in time domain showing Limit Cycle Oscillations (LCO) due to nonlinearities of the flow model and vibrations with large amplitudes. Numerical experiments were performed for the airfoil NACA 0012 respecting the effect of the air space between the flap and the main airfoil.

Mixed Lagrangian–Eulerian description of vortical flows for ideal and viscous fluids

2008 ◽  
Vol 600 ◽  
pp. 167-180 ◽  
Author(s):  
E. A. KUZNETSOV
Keyword(s):  
Stokes Equations ◽  
Diffusion Tensor ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Lagrangian Description ◽  
Navier Stokes Equations ◽  
Vortical Flows ◽  
Eulerian Description ◽  
Vortex Lines

It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid is equivalent to the equations of motion of a charged compressible fluid moving due to a self-consistent electromagnetic field. The velocity of new auxiliary fluid coincides with the velocity component normal to the vorticity line for the primitive equations. Therefore this new hydrodynamics represents hydrodynamics of vortex lines. Their compressibility reveals a new mechanism for three-dimensional incompressible vortical flows connected with breaking (or overturning) of vortex lines which can be considered as one of the variants of collapses. Transition to the Lagrangian description in the new hydrodynamics corresponds, for the original Euler equations, to a mixed Lagrangian–Eulerian description – the vortex line representation (VLR). The Jacobian of this mapping defines the density of vortex lines. It is shown also that application of VLR to the Navier–Stokes equations results in an equation of diffusive type for the Cauchy invariant. The diffusion tensor for this equation is defined by the VLR metric.

scholarly journals NUMERICAL SOLUTION OF FLUID-STRUCTURE INTERACTION PROBLEMS WITH CONSIDERING OF CONTACTS

2021 ◽  
Vol 61 (SI) ◽  
pp. 155-162
Author(s):  
Petr Sváček
Keyword(s):  
Stokes Equations ◽  
Fluid Motion ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Computational Domain ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Computational Mesh ◽  
Arbitrary Lagrangian Eulerian Method ◽  
High Reynolds

This paper is interested in the mathematical modelling of the voice production process. The main attention is on the possible closure of the glottis, which is included in the model with the concept of a fictitious porous media and using the Hertz impact force The time dependent computational domain is treated with the aid of the Arbitrary Lagrangian-Eulerian method and the fluid motion is described by the incompressible Navier-Stokes equations coupled to structural dynamics. In order to overcome the instability caused by the dominating convection due to high Reynolds numbers, stabilization procedures are applied and numerically analyzed for a simplified problem. The possible distortion of the computational mesh is considered. Numerical results are shown.

scholarly journals Simulation of Free Airfoil Vibrations in Incompressible Viscous Flow — Comparison of FEM and FVM

10.14311/1690 ◽  
2012 ◽  
Vol 52 (6) ◽  
Author(s):  
Petr Sváček ◽  
Jaromír Horáček ◽  
Radek Honzátko ◽  
Karel Kozel
Keyword(s):  
Viscous Flow ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Vertical Direction ◽  
Navier Stokes ◽  
Computational Domain ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Incompressible Viscous Flow ◽  
Naca 0012

This paper deals with a numerical solution of the interaction of two-dimensional (2-D) incompressible viscous flow and a vibrating profile NACA 0012 with large amplitudes. The laminar flow is described by the Navier-Stokes equations in the arbitrary Lagrangian-Eulerian form. The profile with two degrees of freedom (2-DOF) can rotate around its elastic axis and oscillate in the vertical direction. Its motion is described by a nonlinear system of two ordinary differential equations. Deformations of the computational domain due to the profile motion are treated by the arbitrary Lagrangian-Eulerianmethod. The finite volume method and the finite element method are applied, and the numerical results are compared.

scholarly journals Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow

2021 ◽  
Vol 11 (11) ◽  
pp. 4748
Author(s):  
Monika Balázsová ◽  
Miloslav Feistauer ◽  
Jaromír Horáček ◽  
Adam Kosík
Keyword(s):  
Compressible Flow ◽  
Nonlinear Elasticity ◽  
Vocal Tract ◽  
Stokes Equations ◽  
Vocal Folds ◽  
Nonlinear Material ◽  
Navier Stokes ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Reliable Solution

This study deals with the development of an accurate, efficient and robust method for the numerical solution of the interaction of compressible flow and nonlinear dynamic elasticity. This problem requires the reliable solution of flow in time-dependent domains and the solution of deformations of elastic bodies formed by several materials with complicated geometry depending on time. In this paper, the fluid–structure interaction (FSI) problem is solved numerically by the space-time discontinuous Galerkin method (STDGM). In the case of compressible flow, we use the compressible Navier–Stokes equations formulated by the arbitrary Lagrangian–Eulerian (ALE) method. The elasticity problem uses the non-stationary formulation of the dynamic system using the St. Venant–Kirchhoff and neo-Hookean models. The STDGM for the nonlinear elasticity is tested on the Hron–Turek benchmark. The main novelty of the study is the numerical simulation of the nonlinear vocal fold vibrations excited by the compressible airflow coming from the trachea to the simplified model of the vocal tract. The computations show that the nonlinear elasticity model of the vocal folds is needed in order to obtain substantially higher accuracy of the computed vocal folds deformation than for the linear elasticity model. Moreover, the numerical simulations showed that the differences between the two considered nonlinear material models are very small.

Iterative method for solving fully-coupled fluid solid systems

2010 ◽  
pp. 653-670
Author(s):  
Elisabeth Longatte
Keyword(s):  
Stokes Equations ◽  
Cross Flow ◽  
Elastic Cylinder ◽  
Navier Stokes ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Ale Formulation ◽  
Volume Method ◽  
Fully Coupled ◽  
Solid System

This work is concerned with the modelling of the interaction of a fluid with a rigid or a flexible elastic cylinder in the presence of axial or cross-flow. A partitioned procedure is involved to perform the computation of the fully-coupled fluid solid system. The fluid flow is governed by the incompressible Navier-Stokes equations and modeled by using a fractional step scheme combined with a co-located finite volume method for space discretisation. The motion of the fluid domain is accounted for by a moving mesh strategy through an Arbitrary Lagrangian-Eulerian (ALE) formulation. Solid dyncamics is modeled by a finite element method in the linear elasticity framework and a fixed point method is used for the fluid solid system computation. In the present work two examples are presented to show the method robustness and efficiency.

Numerical Investigation of Blade Flutter at or Near Stall in Axial Flow Turbomachines

2001 ◽  
Author(s):  
Wolfgang Höhn
Keyword(s):  
Viscous Flow ◽  
Stokes Equations ◽  
Separated Flow ◽  
Parallel Computer ◽  
Navier Stokes ◽  
Computational Domain ◽  
Compressor Cascade ◽  
Governing Equations ◽  
Navier Stokes Equations ◽  
Blade Flutter

During the design of the compressor and turbine stages of today’s aeroengines, aerodynamically induced vibrations become increasingly important since higher blade load and better efficiency are desired. In this paper the development of a method based on the unsteady, compressible Navier-Stokes equations in two dimensions is described in order to study the physics of flutter for unsteady viscous flow around cascaded vibrating blades at stall. The governing equations are solved by a finite difference technique in boundary fitted coordinates. The numerical scheme uses the Advection Upstream Splitting Method to discretize the convective terms and central differences discretizing the viscous terms of the fully non-linear Navier-Stokes equations on a moving H-type mesh. The unsteady governing equations are explicitly and implicitly marched in time in a time-accurate way using a four stage Runge-Kutta scheme on a parallel computer or an implicit scheme of the Beam-Warming type on a single processor. Turbulence is modelled using the Baldwin-Lomax turbulence model. The blade flutter phenomenon is simulated by imposing a harmonic motion on the blade, which consists of harmonic body translation in two directions and a rotation, allowing an interblade phase angle between neighboring blades. Non-reflecting boundary conditions are used for the unsteady analysis at inlet and outlet of the computational domain. The computations are performed on multiple blade passages in order to account for nonlinear effects. A subsonic massively stalled unsteady flow case in a compressor cascade is studied. The results, compared with experiments and the predictions of other researchers, show reasonable agreement for inviscid and viscous flow cases for the investigated flow situations with respect to the Steady and unsteady pressure distribution on the blade in separated flow areas as well as the aeroelastic damping. The results show the applicability of the scheme for stalled flow around cascaded blades. As expected the viscous and inviscid computations show different results in regions where viscous effects are important, i.e. in separated flow areas. In particular, different predictions for inviscid and viscous flow for the aerodynamic damping for the investigated flow cases are found.

scholarly journals PARALLEL COMPUTATIONS IN STUDIES OF DEPENDENCE OF GAS DYNAMIC PARAMETERS OF UPWARD SWIRLING FLOW OF GAS ON BLOWING VELOCITY

2016 ◽  
pp. 92-97
Author(s):  
R. E. Volkov ◽  
A. G. Obukhov
Keyword(s):  
Stokes Equations ◽  
Swirling Flow ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Gas Dynamic

The rectangular parallelepiped explicit difference schemes for the numerical solution of the complete built system of Navier-Stokes equations. These solutions describe the three-dimensional flow of a compressible viscous heat-conducting gas in a rising swirling flows, provided the forces of gravity and Coriolis. This assumes constancy of the coefficient of viscosity and thermal conductivity. The initial conditions are the features that are the exact analytical solution of the complete Navier-Stokes equations. Propose specific boundary conditions under which the upward flow of gas is modeled by blowing through the square hole in the upper surface of the computational domain. A variant of parallelization algorithm for calculating gas dynamic and energy characteristics. The results of calculations of gasdynamic parameters dependency on the speed of the vertical blowing by the time the flow of a steady state flow.

Compressor Blade Optimization Using a Continuous Adjoint Formulation

2006 ◽  
Author(s):  
Dimitiros I. Papadimitriou ◽  
Kyriakos C. Giannakoglou
Keyword(s):  
Stokes Equations ◽  
Compressor Blade ◽  
Navier Stokes ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Adjoint Formulation ◽  
Design Variables ◽  
Gradient Computation ◽  
Geometrical Constraints ◽  
Continuous Adjoint

In this paper, a constrained optimization algorithm is formulated and utilized to improve the aerodynamic performance of a 3D peripheral compressor blade cascade. The cascade efficiency is measured in terms of entropy generation along the developed flowfield, which defines the field objective functional to be minimized. Its gradient with respect to the design variables, which are the coordinates of the Non-Uniform Rational B-Spline (NURBS) control points defining the blade, is computed through a continuous adjoint formulation of the Navier-Stokes equations based on the aforementioned functional. The steepest descent algorithm is used to locate the optimal set of design variables, i.e. the optimal blade shape. In addition to the well-known advantages of the adjoint method, the current formulation has even less CPU cost for the gradient computation as it leads to gradient expression which is free of field variations in geometrical quantities (such as derivatives of interior grid node coordinates with respect to the design variables); the computation of the latter would be costly since it requires remeshing anew the computational domain for each bifurcated design variable. The geometrical constraints, which depend solely on the blade parameterization, are handled by a quadratic penalty method by introducing additional Lagrange multipliers.

Front Tracking Approach and Application to Multi-Fluid Flow

2006 ◽  
Author(s):  
Manasa Ranjan Behera ◽  
K. Murali
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Fixed Grid ◽  
Moving Interface ◽  
Fluid Properties ◽  
Volume Method ◽  
Transfer Of Information

Multiphase flows simulations using a robust interface-tracking method, are presented. The method is based on writing one set of governing equations for the whole computational domain and treating the different phases as single fluid domain with variable material properties. Interfacial terms are accounted for by adding the appropriate sources as δ functions at the boundary separating the phases. The unsteady Navier-Stokes equations are solved by finite volume method on a fixed, structured grid and the interface, or front, is tracked explicitly by a lower dimensional grid. Interfacial source terms are computed on the front and transferred to the fixed grid. Advection of fluid properties such as density and viscosity is done by following the motion of the front. The method has been implemented for interfacial flow problems, depicting the interface and topology change capturing capability. The representation of the moving interface and its dynamic restructuring, as well as the transfer of information between the moving front and the fixed grid, is discussed. Extensions of the method to density stratified flows, and interfacial movements are then presented.

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