A Deterministic Viscous Vortex Method for Grid-free CFD with Moving Boundary Conditions
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1 A Deterministic Viscous Vortex Method for Grid-free CFD with Moving Boundary Conditions M.W. PITMAN, A.D. LUCEY Department of Mechanical Engineering Curtin University of Technology GPO Box U1987, Perth, WA 6845 AURALIA Abstract: - A robust computational method for the grid-free simulation of non-linear, fully-coupled fluidstructure interaction is presented. While the method is robust enough to handle the fluid-structure interaction analysis of non-linearly deforming arbitrarily shaped bodies, the motivation for the development lies in the study of passive boundary layer control using compliant walls. Therefore the method presented in this paper is set in the context of boundary-layer flow interacting with a compliant boundary. The fluid transport equations are solved by a deterministic viscous vortex method based on a corrected core-spreading method. Novel techniques are used to enforce the moving boundary conditions simultaneously at the compliant wall, upstream and downstream far-field boundary conditions, and to evaluate the fluid pressure at the moving boundary. Numerical results are presented to demonstrate the successful implementation for the specific test case of a boundary-layer flow interacting with linear deformations of a compliant surface. Key-Words: - Vortex method, deterministic, core-spreading, fluid-structure interaction, moving boundary conditions. 1 Introduction The present study focuses on the numerical modeling of a finite-length, one-sided compliant surface interacting with a boundary layer flow as shown in Figure 1. It is known, for example see the review of Gad-el-Hak [3], that boundary-layer control can be achieved through wall compliance and the potential for indefinite postponement of laminar-to-turbulent transition exists, thereby yielding skin-friction drag reduction in marine applications. Many theoretical and numerical studies of this system assumed potential flow in order to model divergence, a hydro-elastic instability that must be avoided if compliance is to postpone transition. The research presented builds upon this work by incorporating fully viscous effects in the framework of a robust computational method. The use of a Lagrangian Discrete Vortex Method (DVM) is desirable because it allows efficient modeling at transitional Reynolds numbers without the need for mesh regeneration at each time-step or the implementation of an ad hoc turbulence model as with many grid-based methods. Direct numerical simulation is also too computationally expensive at the typical Reynolds numbers in this problem at hand. The DVM allows for full coupling between the wall and fluid motion. However, solutions obtained through stochastic methods and operator splitting used to implement viscous effects generally suffer from excessive numerical noise, and an inability to enforce rigorously the boundary conditions. Upstream/approaching boundary layer profile Rigid wall upstream Perturbed boundary layer profile Mean-flow U Spring-backed compliant wall section Downstream/exit boundary layer profile Rigid wall downstream Fig.1: One-sided boundary layer flow over a finite compliant wall set in an otherwise rigid wall. In this paper we present a robust solution methodology that is suitable for compliantboundary analysis. A deterministic vortex method that does not rely upon operator splitting is used for the viscous-flow solution and a finite-difference solution for the wall motion. Avoiding operator splitting means that the no-slip and no-flux conditions at the interface may be enforced
2 simultaneously using a boundary element method. Several novel techniques are presented in order to achieve a solution that meets the requirements of being simple to implement numerically, efficient, deterministic and yields a smooth pressure footprint at the wall. Solution Method The fluid transport equations to be solved are the two-dimensional, incompressible Navier-Stokes equations. The curl operator is applied to the equations, which eliminates the pressure variable and produces the vorticity transport equations for the scalar two dimensional vorticity field ω + ( u ) ω = υ ω + ( f ), (1) t where ω is the two-dimensional scalar vorticity field, ν is the kinematic viscosity, u is the two dimensional velocity vector field. f is a forcing function that is zero in the present work. The approximate solution to (1) is generated by a deterministic discrete vortex method based upon the core-spreading method..1 Corrected Core-Spreading Method The vortex method used is the Corrected Core- Spreading Vortex Method (CCSVM) developed by Rossi [6]. Various other viscous vortex methods exist. Other methods considered were the vortex redistribution, random walk and diffusion velocity methods. A review of these methods is found in Takeda [8]. The CCSVM was selected as a desirable viscous vortex method because it allows complete error control, is simple to implement and can be easily merged with a method for implementing upstream/downstream conditions. However, one of the greatest advantages of this method is that the vortex fusion algorithm used to control problem size can also be utilised to detect and deal with near-wall vortices without the need for an explicit algorithm, such as that required if using vortex sheets. The method uses Gaussian smoothed particles (also know as Lamb vortices) that are self-similar solutions to the diffusion equation. The cores of these particles are allowed to spread and diffuse through time exactly according to the solution of the diffusive term. This solves the diffusive term exactly but incorrectly solves for the convective term and causes the uncorrected core-spreading method to approximate the wrong equation. Therefore, the diffusion is corrected by only allowing a vortex to diffuse to a certain limiting size, after which it is split into several more vortex cores and the process is repeated. If this process is continued, the problem size will grow exponentially as more cores are split into many smaller ones. Remediation by a vortex fusion algorithm that merges close vortices into one must therefore be implemented. The processes of vortex splitting and fusion can be error controlled, ensuring consistency of the approximation to the solution throughout.. Wall Motion The linearised equations of motion for the springbacked flexible wall panels with dashpot damping shown in Figure 1 are 4 η η η η ρm. h. + d. + B + K Eη + T = δp, () 4 t t x x where the flexural rigidity is defined by 3 Eh B =, (3) 1( 1 v p ) and where ρ p, ν p, h and E are, respectively the density, Poisson ratio, thickness and elastic modulus of the flexible wall. T is the tension in the plate which for the present work has been set to zero. K E and d are the coefficients of foundationspring stiffness and damping. Finally, δp is the wall forcing function which is taken solely as the fluid pressure at the wall relative to its value in the undisturbed state. The solution of () by a finite-difference method is desirable because of the method s simplicity and the fact that it allows for a natural merging with the boundary-element method that, it will be seen below, is a component of the flow solution. 3 Interfacial Modelling 3.1 Vorticity/Enstrophy Conservation An inherent problem of vortex methods is that as a smoothed vortex element moves close to a wall its core will cross the boundary and vorticity is essentially lost from the flow field that would normally be conserved. This creates problems with determining the flow field close to the boundary. Investigators have attempted various techniques to overcome this problem such as special vortex sheet
3 methods, grid-based solutions and even ignoring the core function for vortices close to the wall. However, these methods tend to produce noisy pressure footprints and suffer from inefficiency in the special handling of near-wall elements Correction for Vorticity below the Wall The near wall flow may be represented by a set of Gaussian vortex sheets that, as with the wall jet study of Rossi [7], approximate the unsteady Rayleigh equation. However, the Gaussian vortex approximation includes vorticity that lies below the wall that does not exist in the solution to the Rayleigh equation. This is cancelled by a point vortex sheet introduced on the wall with half strength and opposite sign to the Gaussian vortex sheet. The resulting boundary element used to impose the no-slip condition is a Gaussian-point vortex sheet couple. An approximate representation of this element using point vortices and Gaussian blobs instead of sheets is shown in Figure. the use of the vortex fusion algorithm to fuse nearwall vortices with the wall vortices. The wall-gaussian cores are able to spread just as the free-vortex cores. At each time step new vortices are introduced into the domain of strength and radius determined by balancing up to the fourth moment of vorticity. After this the wall-cores are returned to their original core-size and the diffusion/spreading-splitting process is repeated in the next time-step. 3. Concurrent Boundary Conditions The most complicated and delicate procedures of any DVM is the handling of boundary conditions near a wall to impose no-slip and no-flux conditions. Utilising source as well as the point- Gaussian vortex couple, these can efficiently be applied simultaneously. A schematic of the wall and near-wall elements that contribute to enforcement of the boundary conditions is shown in Figure. Because core-spreading method does not rely upon operator splitting, these boundary conditions can be applied simultaneously at each time step. Approx. field created by the 3 elements at each subelement Vortex Couple and Source Subelement Wall sources primarily restricts normal velocity flux. Boundary Element Velocities due to various computational elements and the resulting final velocity field Total Velocity Profile Fig.: Schematic of near-wall elements and their approximate influence on the near-wall flow field Implementation For practical implementation, and in order to make use of the efficiencies of a Fast Multipole Method (FMM) for summation, the vortex sheets at each boundary element are represented by a series of point vortices. This is also beneficial for the easy detection of vorticies that have moved within the removal radius of the wall. The wall vorticies in this way behave in a similar manner to the free flow vorticies, except they are constrained in position on the panels. In order to maintain second order spatial accuracy and correct vorticity injection it is important to conserve enstrophy (second moment of vorticity) at the wall. This is achieved through 3..1 The No-Slip Condition A boundary-element Gaussian-point vortex sheet couple is used to enforce the no-slip condition. The influence of each wall Gaussian and point vortex sheet (assuming unit strength) on the other may be represented in matrix form. Using the resulting matrix of influence coefficients, the normal and tangential velocity at the elements due to the wall vortex boundary elements, denoted U and U respectively, are [ I ] [ I ] { γ } = [ I ]{ γ } = { U } G 1 P [ I ] [ I ] { γ } = [ I ]{ γ } = { U } G 1 P, (4.a), (4.b) where γ is a vector defining the strength of each vortex couple boundary element, while I G, I P, I G, and I P are the square influence matrices of tangential and normal velocity respectively due to the Gaussian and point vortices. 3.. The No-Flux Condition A boundary-element source sheet is used to enforce the no-flux condition. By calculating the influence coefficients of each element upon
4 another, the normal and tangential velocity across each element due to the source/sink strengths, denoted U and U respectively, are [ I ]{ } = { } λ, (5.a) U [ I ]{ } = { } λ, (5.b) U where λ is a vector defining the strength of each vortex couple boundary element, while I, I are the square influence matrices of tangential and normal velocity respectively due to the wall sources Simultaneous Enforcement Given the matrix equations represented by the sets of equations (4) and (5). The boundary conditions may be solved simultaneously through the coupled set of matrix equations: [ I ]{ } + [ I ]{ λ} = { U } + { U } = { U } γ, (6.a) [ I ]{ } + [ I ]{ λ} = { U } + { U } = { U } γ. (6.b) Straightforward solution of these equations for the vortex and source strengths yields: 1 { } = ([ I ] + [ K][ I ]) ({ U } + [ K]{ U }) λ, (7.a) 1 { } = ([ I ] + [ K][ I ]) ({ U } + [ K]{ U }) γ, (7.b) for: [ ] 1[ I ][ ] 1 I K. (8) = 4 Far-Field Approximation The upstream and downstream flow conditions are approximated by the use of semi-infinite vortex sheets that are arranged to approximate the Blasius boundary layer profile at the inflow. At the commencement of the simulation the positioning of these vortex sheets is optimised in order to achieve an accurate approximation. The vortex sheets interact with the velocity and vorticity profile through a non-linear function. Each vortex sheet element has three primary parameters associated with it, namely position, strength and core-size that must be optimised. In order to solve this problem a genetic search algorithm is used. Specific details on the stochastic operation of the genetic search algorithm are not discussed in this paper for reasons of brevity. N N T T T N 5 The wall motion The pressure may be determined at each timestep through the method for viscous flows outlined in Lewis [4]. However, as with the numerical work of Davis and Carpenter [], full-coupling with the moving boundary may be achieved if the total unsteady pressure is separated into three terms that correspond to the hydrodynamic stiffness, damping and inertia. We therefore decompose the pressure perturbation as follows. S U1 U δp η, η, η = δp η, η + δp η + δp η. (9) Substituting (9) into the equations of wall motion, (), the acceleration terms may be grouped to one side of the equation and the solution for the wall motion generated by implicit time-marching scheme following the methods of Lucey & Carpenter [5]. 6 Illustrative Results The results presented below are intended to illustrate the optimisation of far-field conditions by the genetic algorithm, the use of the core-spreading method for flow solution and the simultaneous enforcement of boundary conditions. For all results, the boundary layer thickness δ at inflow is m and the mean-flow velocity U is 1 m/s. 6.1 Vorticity Discretisation The results from a typical run of the genetic algorithm (GA) are shown below in Figure 3. Figure 3A shows the best and average values for the scoring function returned by each generation. Figure 3B shows a diagrammatic representation of the vortex sheet positions and core-sizes with labels indicating the strength, number and position of each sheet. Figures 3C and 3D present the vorticity and velocity curves respectively. The dashed line is the desired profile and the solid line is the discrete vortex approximation that was achieved by the genetic algorithm. The solid horizontal lines show the location of the vortex sheet centres for reference. It is important to note that it is the vorticity profile in Figure 3C that the genetic algorithm is actually attempting to approximate with discrete vortices. The velocity profile is then calculated from this result to give the velocity fit shown in Figure 3D.
5 Fig.3 : GA result for the fitting of 10 vortex sheets to the Blasius profile. 6. interface (kpa) interface (kpa) Length from upstream sheets (m) Velocity (m/s) Flow Dynamics Results were presented for both static and moving-wall situations. The purpose of these results is to demonstrate the ability of the method to give a smooth pressure footprint at the wall and qualitatively correct results for the flow field. The properties of the fluid are based upon silicone oil DMS-10, with a density of 950 kg/m3, and a kinematic viscosity of 10-5 m/s. The length of the compliant wall section is m, and the plate thickness of the wall is mm and its flexural rigidity is Nm. The foundation-spring stiffness has been set to zero for the cases presented here Moving-wall Simulation The effect of the fluid on the wall was ignored in these preliminary results, the objective being to demonstrate the application of the core-spreading method to a moving boundary problem. For the case presented, the motion of the wall has been prescribed as the third in-vacuo mode for hinged plate ends. Results for the vortex distribution and pressure footprint for three stages of the simulation are shown below in Figure 5. The three stages were: (A) at commencement of the simulation, (B) just after the first splitting event and (C) much later in the simulation after numerous splitting events. Wall and vortex positions Boundary Element Number (C) Pressure at the wall panels interface (kpa) Vorticity (1/s) Boundary Element Number (B) Pressure at the wall panels Wall and vortex positions Length (m) (D) Velocity fit (C) Vorticity fit (A) Pressure at the wall panels Wall and vortex positions Length from upstream sheets (m) Generation number Scoring function value (A) GA Scoring Function (B) Vortex sheet positions Length from upstream sheets (m) Boundary Element Number Figure 4: Vortex distribution and pressure footprint along the wall at three times within a moving wall simulation: (A) Time step = 14, Time = when number of vortices is 843; (B) Time step = 9, Time = when number of vortices is 6073; and (C) Time step = 95, Time = when number of vortices is 899. The results of Figure (4) demonstrate the ability of the method to restrict vortex proximity to the wall and achieve a smooth pressure distribution even after numerous splitting events have resulted in a random vortex distribution. The simultaneous handling of the no-flux and no-slip boundary conditions at the wall is also demonstrated. 7 Concluding Remarks A new computational method has been developed for modelling boundary-layer flow interactions with a flexible surface. Key features of the method include: a robust corrected core-spreading vortex method that models viscous effects in flow over a compliant boundary. a novel boundary-element method to impose the no-slip and no-flux boundary conditions concurrently at the moving wall-flow interface and with reduced numerical noise.
6 a rapid genetic algorithm that aids the accurate enforcement of upstream and downstream conditions Potential for improvements lie with the use of higher-order accurate corespreading method such as the use of the more recent Elliptical Core Spreading Vortex Method (ECSVM), more efficient Fast Multipole Methods for non-harmonic explicit velocity kernels, efficiency of the vortex merging algorithm, simulated convection of near-wall vortices, All of the techniques developed in the present paper may be extended to model the corresponding three-dimensional system. The finite difference wall code that is currently used only supports linear, one-dimensional motion. Future modelling will include non-linear, two-dimensional motion. The next step for study will be the application of the computational technique to undertake a systematic investigation of hydrodynamic stability, together with travelling-wave flutter and divergence instabilities on finite length compliant panels. Vortex Methods for Engineering Applications, Albuquerque, New Mexico, [8] Takeda, K., Tutty, O.R. and Fitt A.D., 1997, A Comparison of Four Viscous Models for the Discrete Vortex Method. 13th AIAA CFD Conference, Snowmass, Colorado, References: [1] Chorin, A.J., 1980, Vortex Models and Boundary Layer Instability. SIAM Journal on Scientific and Statistical Computing, 1:1-1. [] Davis, C. and Carpenter, P.W., 1997, Numerical Simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels, J. Fluid Mech., 335: pp [3] Gad-El-Hak, M., 003, Drag Reduction using Compliant Walls. In IUTAM Symposium on Flow past Highly Compliant Boundaries and in Collapsible Tubes, P.W. Carpenter and T.J. Pedley (eds.), [4] Lewis, R.I., 1991, Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems, Cambridge University Press. [5] Lucey, A.D. and Carpenter, P.W., 199, A numerical simulation of the interaction of a compliant wall and an inviscid flow J. Fluid Mech., 34: pp [6] Rossi, L.F., 1996, Resurrecting Core Spreading Vortex Methods: A New Scheme that is Both Deterministic and Convergent. SIAM Journal on Scientific Computing, 17: [7] Rossi, L.F., 1994, Vortex Computations of Wall Jet Flows, In Proceedings of Forum on
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