Stephane Etienne

Shape Sensitivity Analysis of Fluid-Structure Interaction Problems

This paper presents a general monolithic formulation for sensitivity analysis of the steady interaction of a viscous incompressible o w with an elastic structure undergoing large displacements (geometric non-linearities). This is a direct extension of our previous work on value parameter sensitivity of such problems. 1 The coupled set of equations is solved in a direct implicit manner using a Newton-Raphson adaptive nite element method. A pseudo-solid formulation is used to manage the deformations of the uid domain. The formulation uses uid velocity, pressure, and pseudo-solid displacements as unknowns in the o w domain and displacements in the structural components. The adaptive formulation is veried on a problem with a closed form solution. It is then applied to sensitivity analysis of three elastic plates placed in a channel o w. Sensitivities are used for fast evaluation of nearby problems (i.e. for nearby values of the parameters or geometric characteristics) and for cascading uncertainty through the Computational Fluid Dynamics/Computational Structural Dynamics code to yield uncertainty estimates of the deformed plates shape.

Sensitivity Analysis of Unsteady Fluid-Structure Interaction Problems

´This paper presents a general monolithic formulation for sensitivity analysis of the unsteady interaction of a viscous incompressible flow with an elastic structure undergoing large displacements (geometric non-linearities). This is a direct extension of our previous work on value parameter sensitivity of such problems. 1,2 The coupled set of equations is solved in a direct implicit manner using a Newton-Raphson finite element method. A pseudo-solid formulation is used to manage the deformations of the fluid domain. The formulation uses fluid velocity, pressure, and pseudo-solid displacements as unknowns in the flow domain and displacements in the structural components. The finite element method is verified on a problem with a closed form solution. It is then applied to sensitivity analysis of an elastic plate placed in a channel flow. Sensitivities are used for fast evaluation of nearby problems (i.e. for nearby values of the parameters or geometric characteristics) I. Introduction This paper presents a formulation suitable for simulating the interaction between an incompressible flow and a structure undergoing large displacements and for computing its sensitivities with respect to parameters of interest. We assume existence and uniqueness of the solution. Previous works have been published on sensitivity analysis of Fluid-Structure Interactions (FSI) 3‐7 but not with the continuous sensitivity equation (CSE).

A Second-order sensitivity equation method for laminar flow

This paper presents a general Continuous Sensitivity Equation (CSE) method for computing first and second-order flow sensitivities for the incompressible Navier–Stokes equations. The sensitivity equations and boundary conditions are developed for value parameters that do not affect the geometry of the computational domain. Applications are thus restricted to the simpler case of value parameters. The flow and sensitivity equations are solved by an adaptive finite element method. The proposed methodology is verified on a problem with a closed form solution. The verified code is then applied to compute first- and second-order flow sensitivities of an airfoil with respect to the angle of attack and the free-stream velocity. We demonstrate the use of sensitivities for fast first- and second-order evaluations of nearby flows. The methodology is also used to perform second-order uncertainty analysis.

Code Verification and the Method of Manufactured Solutions for Fluid-Structure Interaction Problems

This paper presents the Method of Manufactured Solutions (MMS) for uid-structure interactions code verication. The MMS provides benchmark solutions for direct evaluation of the solution error. The best benchmarks are exact analytical solutions with sucien tly complex solution structure to ensure that all terms of the dieren tial equations are exercised in the simulation. The MMS provides a straight forward and general procedure for generating such solutions. When used with systematic grid renemen t studies, which are remarkably sensitive, the MMS provides strong code verication with a theorem-like quality. The MMS is rst presented on simple 1-D examples. Manufactured solutions for uid-structure interaction (FSI) problems are then presented with sample results from grid convergence studies.

The sensitivity equation method in fluid mechanics

We present the sensitivity Equation Method (SEM) as a complementary tool to adjoint based optimisation methods. Flow sensitivities exist independently of a design problem and can be used in several non-optimization ways: characterization of complex flows, fast evaluation of flows on nearby geometries, and input data uncertainties cascade through the CFD code to yield uncertainty estimates of the flow field. The Navier-Stokes and sensitivity equationssensitivity are solved by an adaptive finite element method.

Moderator Flow Simulation Around Calandria Tubes of Candu-6 Nuclear Reactors

Abstract CFD simulations of cross-flows along in-line and staggered tube bundles which emulate those encountered in the calandria of CANDU-6 reactors are presented. The knowledge of external wall temperature distributions around calandria tubes is a major concern during normal and off-normal operating conditions of CANDU reactors. Calculations are performed using the FLUENT software with several turbulence models using segregated and Coupled algorithms. It is observed that κ-based models are able to reproduce mean velocities in staggered bundles. In most cases, the Coupled algorithm yields convergence even if it requires a longer computational time. Based on this work, the standard κ-ε model is recommended to perform this kind of simulations. Improved κ-ε models do not lead to better results while the κ-ω model predicts very well the physics only around the first row but it is unable to predict the flow around tubes located far downstream in the bundle.

hp-Adaptive time integration based on the BDF for viscous flows

This paper presents a procedure based on the Backward Differentiation Formulas of order 1 to 5 to obtain efficient time integration of the incompressible Navier-Stokes equations. The adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process. The stepsize selection (h-adaptivity) is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance. The order selection (p-adaptivity) relies on the idea that low-accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low. The selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize. Hence, it guarantees both that (1) the used method of integration operates inside of its stability region and (2) the time integration procedure is computationally efficient. The proposed time integration procedure also features a time-step rejection and quarantine mechanisms, a modified Newton method with a predictor and dense output techniques to compute solution at off-step points.

Experimental Study on the Flow Around Two Tandem Cylinders with Unequal Diameters

In this paper, flow around two circular cylinders in tandem arrangement with unequal diameters has been investigated using the particle image velocimetry technique (PIV) in a water channel. The upstream to downstream diameter ratio was kept constant at d/D = 2/3, the centre-to-centre distance was varied from 1.2D to 5D and the Reynolds number was varied from 1200 to 4800. The flow characteristics were analyzed through ensemble-averaged patterns of velocity, vorticity, normalized Reynolds stress contours and streamlines. Based on ensemble-averaged and instantaneous flow fields, different flow patterns, including single-wake-shedding at small spacing ratio, bi-stable flow behavior (alternating behavior of reattachment and vortex shedding) at intermediate spacing ratio and co-shedding pattern at large spacing ratio were observed. The effects of Reynolds number and the centre-to-centre spacing ratio on flow patterns and turbulent characteristics were also investigated. It was found that the diameter ratio appears to have a certain effect on the flow patterns at intermediate spacing ratios, where the reattachment of shear layer depends on the lateral width of the wake flow in the lee of the upstream cylinder. Extensive discussion on the distributions of Reynolds stress and turbulent kinetic energy was presented.

A continuous Lagrangian sensitivity equation method for incompressible flow

A continuous Lagrangian sensitivity equation method (CLSEM) is presented as a cost effective alternative to the continuous (Eulerian) sensitivity equation method (CESEM) in the case of shape parameters. Boundary conditions for the CLSEM are simpler than those of the CESEM. However a mapping must be introduced to relate the undeformed and deformed configurations thus making the PDEs more complicated. We propose the use of pseudo-elasticity equations to provide a general framework to generate this mapping for unstructured meshes on complex geometries. The methodology is presented in details for the incompressible Navier-Stokes and sensitivity equations in variational form. The PDEs are solved with an adaptive FEM. Sensitivity data obtained with both approaches for a flow around a NACA 4512 are used to obtain estimates of flows around nearby geometries. Results indicate that the CLSEM produces significant improvements in terms of both accuracy and CPU time.

Time-integration for incompressible viscous flows: Stepsize and order selection based on the BDF

This paper presents a procedure based on the Backward Differentiation Formulas to obtain efficient time integration of the incompressible Navier-Stokes equations. The adaptive algorithm performs both stepsize and order selection to control respectively the solution accuracy and the computational efficiency of the time integration process. The stepsize selection (h-adaptivity) is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance. The order selection (p-adaptivity) relies on the idea that low-accuracy solutions can be computed efficiently by low order time integrators while highly accurate solutions require high order time integrators to keep computational time low. Hence, the algorithm selects the most appropriate method within the formulas of order 1 to 5 based on the prescribed solution accuracy and equation stiffness. Doing so it guarantees that the variable stepsize BDF methods used always are stable during the whole time integration interval. The adaptive algorithm behaviors and performances are illustrated on the flow over a circular cylinder at low Reynolds numbers.