Interaction of Incompressible Fluid and Moving Bodies

Size: px

Start display at page:

Download "Interaction of Incompressible Fluid and Moving Bodies"

Ernest Ryan
5 years ago
Views:

1 WDS'06 Proceedings of Contributed Papers, Part I, 53 58, ISBN MATFYZPRESS Interaction of Incompressible Fluid and Moving Bodies M. Růžička Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The subject of this article is the numerical simulation of the interaction of two-dimensional incompressible viscous fluid and a vibrating airfoil. A solid airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the finite element solution of the Navier-Stokes equations coupled with a system of ordinary differential equations describing the airfoil motion. The time-dependent computational domain and a moving grid are taken into account with the aid of the Arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations. High Reynolds numbers up to 10 6 require the application of a suitable stabilization of the finite element discretization. Numerical tests prove that the developed method is sufficiently accurate and robust. Introduction The interaction of the fluid and bodies plays a significant role in many branches of engineering. Example includes air plane industry, where the point is to observe and study deformations of wings induced by flowing air. In this article we are concerned with numerical solution of an aero-elastic problem of two dimensional viscous incompressible flow over an air profile with two degrees of freedom in a wind tunnel. In our model the airfoil is represented by a solid body, which can perform vertical and torsional vibrations. Mathematical model of the flow is formed by the system of two-dimensional non-stationary Navier-Stokes equations and the continuity equation, coupled by initial and mixed boundary conditions. Stabilized finite element method is used to get a numerical solution. With regard to the moving airfoil the computational domain is time-dependent. This involvement requires to use a suitable technique for the simulation on a moving computational grid. To solve this we choose Arbitrary Lagrangian-Eulerian method. In terms of this technique the Navier-Stokes equations are reformulated. In a proper accuracy the used finite element method leads to a large discrete system of nonlinear algebraic equations. We solve the nonlinearity by implementing the Oseen method. This method splits the nonlinearity into a sequence of linear problems. To capture the solution of the large linear system we use direct solvers working sufficiently up to 10 5 unknowns. For a larger system it is necessary to implement iterative methods or another sophisticated methods as Domain Decomposition or Multigrid method. The computed results have been compared with experimental data, obtained in the Institute of Thermomechanics, Academy of Sciences of the Czech Republic. Mathematical model The two-dimensional non-stationary flow of viscous, incompressible fluid is considered in the time interval [0, T], where T > 0. The symbol Ω t denotes the computational domain occupied by the fluid at time t. The boundary Ω t = Γ D Γ O Γ Wt, where the sets Γ D, Γ O a Γ Wt are mutually disjoint and boundary conditions of different types are used there. The symbol Γ D represents the inlet, where the fluid flows into the domain Ω t. The fixed, impermeable wall, Γ O represents the outlet, where the fluid flows out and Γ Wt is the boundary of the profile at the time t. Let us assume that Γ D is independent of time in contrast to Γ Wt. The flow is characterized by the velocity field u = u(x, t), and the kinematic pressure p = p(x, t), for x Ω t and t [0, T]. The kinematic pressure is defined as P/ρ, where P is the pressure and ρ = const. > 0 is the density of the fluid. The aim is to find functions α(t) and h(t), describing the oscillation of the air profile (rotation and drift). The shape of the domain Ω t depends on functions α(t) and h(t) as shown on Figure 1., where TR represents the elastic axis, which can move along the line p. 53

2 Fig. 1. Model scheme. ALE formulation of Navier-Stokes equations ALE (Arbitrary Lagrangian Eulerian) description is given by a smooth, one-to-one mapping A t : Ω 0 Ω t, X x(x, t) = A t (X). (1) For each time t I = [0, T] A t represents a smooth mapping of the reference domain Ω 0 onto the domain Ω t, which is identical with the reference domain Ω 0 in the part of its boundary, where there is no interaction with the body and also there is no deformation of the boundary. The reference domain Ω 0 is identical with the domain occupied by the fluid at initial time t = 0. The coordinates of points x Ω t are called spatial coordinates, the coordinates of points X Ω 0 are called ALE coordinates or reference coordinates. First, we define the domain velocity in the following way This velocity can be expressed in spatial coordinates as w(t, X) = x(x, t). (2) t w = w(x, t) A 1 t, so w(x, t) = w ( A 1 t (x), t ). (3) Let us consider a function f = f(x, t), x Ω t, t [0, T], f(x, t) IR, where IR is the set of real numbers. Let f(x, t) = f(a t (X), t). We can define the ALE derivative of the function f by D A Dt The application of the chain rule gives f f(x, t) = (X, t), t X = A 1 t (x). (4) D A Dt f = f + w f. (5) t By using this relation we can obtain the Navier-Stokes equations in the form D A Dt u + [(u w) ] u + p ν u = 0 in Ω t div u = 0 in Ω t. (6) The symbol ν means the kinematic viscosity of the fluid. We assume that ν > 0 is constant. Equations for the moving air profile The equations for the moving profile are derived from the Lagrange equations for the generalized coordinates h and α. In the matrix calculus these equations have the form Kd(t) + Bḋ(t) + M d(t) = f(t), (7) 54

3 where the stiffness matrix K, the viscous damping B and the mass matrix M have the form ( ) ( ) ( ) khh k K = hα Dhh D, B = hα m Sα, M = k αh k αα D αh D αα S α I α and the vector of the force f and the generalized coordinates d read ( ) ( ) L2 (t) h(t) f(t) =, d =. M(t) α(t) The symbol L 2 stands for the component of the force acting on the profile in the vertical direction, M is the torsional moment of force, D hh, D hα, D αh, D αα are the coefficients of damping, S α, I α, m and k hh, k αα, k hα, k αh denote in order the static moment around the elastic axis TR, the moment of inertia around TR, the mass of the profile and the stiffness of the profile. The functions L 2 and M acting on the profile with a width H are given by the relations 2 L 2 = T 2j n j ds H, (8) M = Γ Wt j=1 Γ Wt i,j=1 2 T ij n j ( 1) i ( x 1+δ1i x TR ) 1+δ 1i ds H, (9) where n is the unit outer normal to Ω t on Γ Wt, and δ ij is the Kronecker symbol, i.e., δ ij = 1 for i = j and δ ij = 0 for i j, x 1, x 2 are the coordinates of points on Γ Wt, which lay on the border of the profile, x TR i, i = 1, 2, are the coordinates of the elastic axe x TR and [ ( ui T ij = ρ pδ ij + ν + u )] j. (10) x j x i Initial and boundary conditions The Navier-Stokes equations must be completed by the initial condition u(x, 0) = u 0, x Ω 0, (11) and the boundary conditions. The part of the boundary Γ D represents the inlet and the impermeable fixed wall. On this boundary we specify the Dirichlet boundary condition u ΓD = u D. (12) The part of the boundary Γ O is the outlet, where we define the so-called do-nothing boundary condition (p p ref )n + ν u n = 0 on Γ O, (13) where p ref is a given reference pressure. On Γ Wt we consider the condition Moreover we define the initial conditions u ΓWt = ũ Wt = w ΓWt. (14) α(0) = α 0, α(0) = α 1, where α 0, α 1, h 0, h 1 are input parameters of the model. h(0) = h 0, ḣ(0) = h 1, (15) 55

4 Discretization of the problem Discretization in time We use equidistant partition of the time interval [0, T], formed by 0 = t 0 < t 1 < < T, t k = kτ, where τ > 0 is a used time step. On each time level we obtain the problem to find functions u n+1 : Ω tn+1 IR 2 and p n+1 : Ω tn+1 IR such that 3u n+1 4û n + û n 1 + ( (u n+1 w n+1 ) ) u n+1 ν u n+1 + p n+1 = 0 v Ω tn+1, 2τ div u n+1 = 0 v Ω tn+1. (16) This system is considered with the boundary conditions (12),(13), (14). The symbols û n and û n 1 mean the functions u n and u n 1 transformed from the domain Ω tn and Ω tn 1 to the domain Ω tn+1 using the ALE mapping. Discretization in space The starting point for the approximate solution is the weak formulation of problem (16). For this purpose the appropriate function spaces are used W = (H 1 (Ω)) 2, X = {v W; v ΓD Γ Wt = 0} and M = L 2 (Ω). We introduce the notation where and a(u, U, V ) = 3 2τ (u, v) + ν((u, v)) + (((u w n+1 ) )u, v) f(v ) = 1 2τ (p, v) + ( u, q), (17) ( ) 4û n û n 1, v v nds, Γ O (a, b) = Ω ab dx U = (u, p) W M U = (u, p) W M V = (v, q) X M. The solution of the weak formulation is U = (u, p), such that it satisfies the conditions U W M, a(u, U, V ) = f(v ), V = (v, q) X M, (18) and u satisfies the boundary conditions (12) and (14). Now we define an approximate solution. We approximate the spaces W, X, M by their finite dimensional subspaces W, X, M, (0, 0 ), 0 >0, where X = { v W ; v ΓD Γ Wt = 0 }. This means that for each (0, 0 ) we assign finite dimensional subspaces W, X, M, with dimensions dimw = n W ( ), dim X = n X ( ), dimm = n M ( ). The approximate solutions is defined as a couple U = (u, p ) W M such that a(u, U, V ) = f(v ), V = (v, q ) X M (19) and u satisfies a suitable approximation of the boundary conditions (12) a (14). The spaces of finite elements X a M must satisfy the Babuška-Brezzi (BB) condition, which guarantees the stability of the used scheme. Stabilization of the finite element method In order to get good results, it is necessary to apply the streamline diffusion stabilization of the finite element method given in [Lube G.]. The reason is that the standard application of the finite element method leads to numerical schemes that give non-physical results represented by non-physical oscillations, especially for high Reynolds numbers. This phenomenon is called Gibb s oscillations. 56

5 We have the triangulation T of the domain Ω t with triangles K i. We define the stabilization components there ( ) 3 L (U, U, V ) = δ K u ν u + (w )u + p, (w )v (20) 2τ K i T K i ( ) 1 F (V ) = 2τ (4ûn û n 1 ), (w )v K i T δ K where U = (u, p) U = (u, p) V = (v, q), δ Ki 0 are suitable parameters, w = u w n+1 is the transport velocity and (, ) Ki is the scalar product in space L 2 (K i ). Next, we define the stabilization component for the pressure in the form P (U, V ) = τ Ki ( u, v) Ki, U = (u, p) V = (v, q), (21) K i T with suitable parameters τ Ki 0. The solution of the stabilized discrete problem is U = (u, p ) W M, such that the component u satisfies the boundary conditions (12) in Γ D and (14) in Γ Wt and a (U, U, V ) + L (U, U, V ) + P (U, V ) = f (V ) + F (V ), (22) V = (v, q ) X M. K i, Now, we introduce the way how to choose parameters δ K and τ K. The magnitude of the velocity field varies in different sub-domains of Ω 0. That is why, we split the domain into two sub-domains. The diffusion component dominates on the first sub-domain and the convective component on the second. On both sub-domains we choose these parameters in a different way. The parameter δ K is based on the transport velocity w and the viscosity ν. We put K δ K = δ ξ(re w ), (23) 2 w L (K) where Re w = δ K w L (K) (24) 2ν is the so-called local Reynolds number and K is size of the element K measured in the direction of w. The function ξ( ) is non-decreasing in dependence on Re w in such a way, that for local convective dominance (Re w > 1) ξ 1 and for local diffusion dominance (Re w < 1) ξ 0. The parameter δ (0, 1] is chosen suitably. The function ξ( ) could be defined e.g. ( ξ(re w Re w ) ) = min 6, 1. (25) The choice of the parameter τ K is different in local convective dominance sub-domain and in local diffusion dominance sub-domain. τ K = τ K w L (K) and τ K = 0 (26) for local convective dominance, respectively local diffusion dominance. The symbol τ means a parameter from (0, 1]. Numerical solution The triangulation of the domain is realized by the software ANGENER, which can be used for initial isotropic triangulation and also for anisotropic adaptive refinement. We use this technique for obtaining the best performance/accuracy rate in the computational mesh (see Figure 2.). As a result we obtain the pressure and the velocity fields and also the position of the moving profile. From this information we derive frequency characteristics, which is also the output of experiments. This fact gives us the possibility to compare both approaches. 57

6 Fig. 2. Anisotropically adapted mesh. main frequency [Hz] f 1 experiment f 2 experiment f 1 numerical experiment f 2 numerical experiment velocity [m/s] Conclusion Fig. 3. Resonance frequency. In this article we derived a procedure for obtaining the numerical solution of the interaction of the moving airfoil with the fluid. We used this approach for solving a particular problem, which was also studied experimentally. On several examples it is shown that the presented method is sufficiently robust and useful for a given type of problem. We show the comparison of the computed and experimentally obtained resonance frequency of induced vibration of the profile for several values of the initial velocity (see Figure 3). For improvement of the accuracy of the computation we could implement a turbulent model to the description of the flow. The further goal is to verify the method on the other test problems, mainly for a higher deviation of the profile and a higher speed of the air, when the system may loose its stability. Acknowledgments. This research was partially supported by grant number project MPO TAN- DEM subproject FT-TA/026, number OV , part 13. and grant number 344/2005/B-MAT/MFF of the Grant Agency of Charles University. References Dolejší V.: ANGENER V3.0, dolejsi/angen/angen.htm, Matematicko-fyzikální fakulta Univerzity Karlovy. Horáček J.: Nelineární formulace kmitání profilu pro aero-hydroelastické výpočty. In: Dynamika strojů 2003, Praha , ÚT AVČR, Praha, 2003, [ISBN ] Horáček J., Kozánek J., Veselý J.: Dynamic and stability properties of an aeroelastic model. In: Inženýrská mechanika 2005, Svratka , ÚT AVČR, Praha 2005, (CD ROM, 12 str.). [ISBN ] Horáček J., Luxa M., Vaněk F., Veselý J., Vlček V.: Návrh experimentálního zařízení pro studium nestacionárních 2D aeroelastických jevů optickými metodami, Výzkumná zpráva Ústavu termomechaniky AV ČR č. Z 1347/04, listopad Lube G.: Stabilized Galerkin finite element methods for convection dominated and incompressible flow problems, Num. Anal. and Math. Model., Banach Center publications (29), Warszawa,

NUMERICAL SIMULATION OF INTERACTION BETWEEN INCOMPRESSIBLE FLOW AND AN ELASTIC WALL

Proceedings of ALGORITMY 212 pp. 29 218 NUMERICAL SIMULATION OF INTERACTION BETWEEN INCOMPRESSIBLE FLOW AND AN ELASTIC WALL MARTIN HADRAVA, MILOSLAV FEISTAUER, AND PETR SVÁČEK Abstract. The present paper