An Immersed Interface Method for the Incompressible Navier-Stokes Equations in Irregular Domains

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1 An Immersed Interace Method or the Incompressible Navier-Stokes Equations in Irregular Domains Duc-Vinh Le, Boo Cheong Khoo, Jaime Peraire Singapore-MIT Alliance Department o Mechanical Engineering, National University o Singapore Department o Aeronautics and Astronautics, Massachusetts Institute o Technology Abstract We present an immersed interace method or the incompressible Navier Stokes equations capable o handling rigid immersed boundaries. The immersed boundary is represented by a set o Lagrangian control points. In order to guarantee that the no-slip condition on the boundary is satisied, singular orces are applied on the luid at the immersed boundary. The orces are related to the jumps in pressure and the jumps in the derivatives o both pressure and velocity, and are interpolated using cubic splines. The strength o singular orces is determined by solving a small system o equations at each time step. The Navier-Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method or pressure and velocity. Keywords: Immersed interace method, Navier-Stokes equations, Cartesian grid method, inite dierence, ast Poisson solvers, irregular domains. I. INTRODUCTION This paper considers the immersed interace method (IIM) or the incompressible Navier-Stokes equations in general domains involving rigid boundaries. In a 2-dimensional bounded domain Ω that contains a rigid interace Γ, we consider the incompressible Navier-Stokes equations, written as u t + (u )u + p = µ u + F (1) with boundary and initial conditions u = (2) u Ω = u b (3) u(x, ) = u (4) where u is the luid velocity, p the pressure, and µ the viscosity o the luid. Here, we simply assume that the density, ρ 1, and the viscosity, µ, are constant. The singular orce F has the orm F (x, t) = (s, t)δ(x X(s, t))ds (5) Γ where X(s, t) is the arc-length parameterization o Γ, s is the arc-length, x = (x, y) is spatial position, and (s, t) is the orce density. The Navier-Stokes equations are discretized using inite dierences on a staggered Cartesian grid. The main eatures o our method are: It is a Cartesian grid method; the method does not require complex mesh generation. It is second order accurate or velocities. The Poisson-like equations resulting at each time step are solved using a cyclic reduction algorithm which has a complexity O(N logn), where N is the number o degrees o reedom. Methods utilizing a Cartesian grid or solving interace problems or problems with complex geometry have become popular in recent years. One o the most successul Cartesian grid methods is Peskin s immersed boundary (IB) method ( [1], [11], [15]). In order to deal with rigid boundaries, Lai and Peskin [11] propose to evaluate the orce density using an expression o the orm, (s, t) = κ(x e (s) X(s, t)), (6) where κ is a constant, κ 1, and X e is the arc-length parameterization o the required boundary position. The orcing term in Eq (6) is a particular case o the eedback orcing ormulation proposed by Goldstein et al. [12] with β =. In [12], the orce is expressed as (s, t) = α t U(s, t )dt + βu(s, t) (7) where U is the velocity at the control points, and α and β are chosen to be negative and large enough so that U will stay close to zero. Lima E Silva et al. [15] proposed an alternative model to compute the orce density based upon the evaluation o the various terms in the momentum equation (1) at the control points. The orce density is calculated by computing all the Navier-Stokes terms at the control points. Once the orce density is obtained at the boundary, the immersed boundary method uses a discrete delta unction to spread the orce density to the nearby Cartesian grid points. Since the IB method uses the discrete delta unction approach, it smears out sharp interace to a thickness o order o the meshwidth and it is only irst-order accurate or problems with non-smooth but continuous solutions. In contrast, the immersed interace method (IIM) can avoid this smearing and maintains a second-order accuracy by incorporating the known jumps into the inite dierence

2 scheme near the interace. The IIM was originally proposed by LeVeque and Li [3] or solving elliptic equations, and later extended to Stokes lows by LeVeque et al. [4]. The method was developed urther or the Navier-Stokes equations in Li et al. [6], Lee [9] and Le et al. [16] or problems with lexible boundaries. The method was also used by Calhoun [8] and Li et al. [7] or solving the two-dimensional streamunctionvorticity equations in irregular domains. In [7], [8] the no-slip boundary conditions are imposed directly by determining the correct jump conditions or streamunction and vorticity. Another Cartesian grid approach has been presented by Ye et al. [13] and Udaykumar et al. [14] using inite volume techniques. They reshaped the immersed boundary cells and use a polynomial interpolating unction to approximate the luxes and gradients on the aces o the boundary cells while preserving second-order accuracy. In this paper, we extend our earlier works presented in Le et al. [16] or problems with deormable boundaries, to solve problems with rigid immersed boundaries. Our approach uses the immersed interace method to solve the incompressible Navier-Stokes equations ormulated in primitive variables. In [16], the singular orce is computed based on the coniguration o the interace, i.e., the interace is assumed to be governed by either surace tension, or by an elastic membrane. In present work, the singular orce at the immersed boundary is determined to impose the no-slip boundary condition at the rigid boundary. At each time step the singular orce is computed implicitly by solving a small, dense linear system o equations. Having computed the singular orce, we then compute the jump in pressure and jumps in the derivatives o both pressure and velocity. The jumps in the solution and its derivatives are incorporated in the inite dierence discretization to obtain sharp interace resolution. Fast solvers rom Fishpack sotware library [18] are used to solve the resulting discrete systems o equations. The remainder o the paper is organized as ollows. In section II, we present the relations that must be satisied along the immersed boundary between the singular orce and the jumps in the velocity and pressure and their derivatives. In section III, we describe the generalized inite dierence approximations to the solution derivatives, which incorporate solution jumps. In section IV, we present details o the numerical algorithm. In section V, some numerical examples are presented and inally, some conclusions and suggestions or uture work are given in section VI. II. JUMP CONDITIONS ACROSS THE INTERFACE Let n and τ be the unit outward normal and tangential vectors to the interace, respectively. The normal, 1 = (s, t) n, and tangential, 2 = (s, t) τ, components o the orce density, can be related to the jump conditions or pressure and velocity as ollows (see [4], [6], [9] or details), [u] =, [µu ξ ] = 2 τ, [u η ] = (8) [p] = 1, [p ξ ] = 2 s, [p η] = 1 s. (9) [µu ηη ] = κ 2 τ, [µu ξη ] = 2 η τ κ 2n, [µu ξξ ] = [µu ηη ] + [p ξ ]n + [p η ]τ + [u ξ ]u n (1) The jump, [ ], denotes the dierence between the value o its argument outside and inside the interace, and (ξ, η) are the coordinates associated with the directions o n and τ, respectively. Here, κ is the signed valued o the curvature o the interace (i.e. we assume that n τ = k constant, so that n can point either towards, or outwards rom, the center o curvature). From expressions (8) (1) the values o the jumps o the irst and second derivatives o velocity and pressure with respect to the (x, y) coordinates are easily obtained by a simple coordinate transormation. For instance, we write, [u x ] = [u ξ ]n 1 + [u η ]τ 1 [u yy ] = [u ξξ ]n [u ξη ]n 2 τ 2 + [u ξη ]τ2 2, where n = (n 1, n 2 ) and τ = (τ 1, τ 2 ) are the Cartesian components o the normal and tangential vectors to the interace at the point considered. III. GENERALIZED FINITE DIFFERENCE FORMULAS From Taylor series expansions, it is possible to show that i the interace cuts a grid line between two grid points at x = α, x i α < x i+1, then, the ollowing approximations hold or a piecewise twice dierentiable unction v(x): v x (x i ) = v i+1 v i 1 v x (x i+1 ) = v i+2 v i (h + ) m [v (m) ] + O(h 2 ) m= 2 (h ) m [v (m) ] + O(h 2 ) m= v xx (x i ) = v i+1 2v i + v i 1 h 2 1 h 2 v xx (x i+1 ) = v i+2 2v i+1 + v i h h 2 2 (h + ) m [v (m) ] + O(h) m= 2 (h ) m [v (m) ]+O(h) m= where v (m), denotes the m-th derivative o v, v i = v(x i ), h + = x i+1 α, h = x i α, and h, is the mesh width in the x direction. The jumps in v and its derivatives are deined as [v (m) ] α = lim x α + v(m) (x) lim x α v(m) (11) in short, [ ] = [ ] α, and v () Bube [5] or more details. A. Projection method IV. NUMERICAL ALGORITHM = v. See Wiegmann and We employ a pressure-increment projection algorithm or the discretization o the Navier-Stokes equations. This projection algorithm is analogous to that presented in Brown et al. [1]. It leads to a second order accuracy or both velocity and pressure provided all the spatial derivatives are approximated to second order accuracy. The spatial discretization is carried out on a

3 standard MAC staggered grid analogous to that in Kim and Moin [2]. The ENO second-order upwind scheme is used or the advective terms (Shu and Osher [17]). With the MAC mesh, the pressure ield is deined at the cell center where the continuity equation is enorced. The velocity ields u and v are deined at the vertical edges and horizontal edges, respectively. Given the velocity u n, and the pressure p n 1/2, we compute the velocity u n+1 and pressure p n+1/2 in three steps: Step 1: Compute an intermediate velocity ield u by solving u u n = (u u) n+ 1 2 p n µ 2 u n+ 1 2 (12) u Ω = u n+1 b where, the advective term is extrapolated using the ormula, (u. u) n+ 1 2 = 3 2 (u u)n 1 2 (u u)n 1 (13) the diusion term is approximated implicitly as, 2 u n+1/2 = 1 2 ( 2 hu + 2 hu n ) + C 1. (14) and the pressure gradient term is given by p n 1 2 = G MAC p n C2 (15) The MAC gradient operators are deined as (G MAC x p) i+ 1 2,j = p i+1,j p i,j x (G MAC y p) i,j+ 1 2 = p i,j+1 p i,j y Step 2: Compute a pressure update φ n+1 by solving the Poisson equation 2 φ n+1 = u, n φ n+1 Ω = (16) This is accomplished by solving the discrete system 2 hφ n+1 = DMAC u + C 3 (17) where the MAC divergence operator is deined as ollows (D MAC u) i,j = u i+ 1 2,j u i 1 2,j + v i,j+ 1 v 2 i,j 1 2 x y Step 3: Update pressure and velocity ield according to, u n+1 = u tg MAC φ n+1 + C 4 (18) p n+1/2 = p n 1/2 + φ n+1 µ ( D MAC u ) + C 5 (19) 2 The operators h and 2 h are the standard three point central dierence operators and C i, i = 1,..., 5, are the correction terms which are only non-zero at the points near the interace and are calculated using generalized inite dierence ormulas o the type introduced in the previous section. This method requires solving two Helmholtz equations or u in Eq (12) and one Poisson equation or φ n+1 in Eq (17). Since the correction terms only aect the right-hand sides o the discrete systems, we can take advantage o the ast solvers rom Fishpack [18] to solve these equations. B. Singular orce evaluation Having solved or u n+1 at the grid points, we now compute the velocity at the interace. In our method, we use a set o control points to represent the interace. The velocity at the control points, U k, is interpolated rom the velocity at the grid points. Thus, we can write U k = U(X k ) = B(u n+1 ) (2) where B is the bilinear interpolation operator which includes the appropriate correction terms which are required to guarantee second order accuracy when the derivatives o the velocity are discontinuous. In summary, the equations that need to be solved in order to calculate u n+1 and U k, can be written symbolically as, Eq (12) Hu = C + B 1 Eq (17) Lφ n+1 = Du + B 2 Eq (18) u n+1 = u Gφ n+1 + B 3 Eq (2) U k = Mu n+1 + B 4 Eliminating u, φ n+1 and u n+1 rom the above equations, we can compute the velocity U k at the control points as ollows, U k = M ( H 1 C GL 1 DH 1 C ) + (M ( H 1 B 1 GH 1 B 1 GL 1 ) ) B 2 + B 3 + B4 For convenience, we can write the above equation as U k = U k + A (21) where U k is simply the velocity at the control points obtained by solving Eqs (12) (2) with =, given u n and p n 1/2. A is a 2N b 2N b matrix, where N b is the number o control points. The vector A is the velocity at the control points obtained by solving the ollowing equations u = µ 2 2 u, u Ω = (22) 2 φ n+1 = u, n φ n+1 Ω = (23) u n+1 = u t φ n+1 (24) A = B(u n+1 ) (25) with being the singular orce at the immersed boundary. Eq (21) can be used to determine the singular orce i we know the prescribed velocity U p at the immersed boundary. Thus, the singular orce at the control points can be computed by solving A = U p U k (26) The matrix A is computed once and stored. We solve Eqs (22) (25) 2N b times, i.e., one or each column. Each time, the orce strength is set to zero except or the entry in the column we want to calculate which is set to one. Once the matrix A has been calculated, only the right hand side U p U k, needs to be computed at each timestep. The resulting small system o equations (26) is then solved at each timestep or the singular orce. Finally, we solve Eqs (12) (19) to obtain u n+1 and p n+1/2.

4 Z U ield at t = 1 N N b E(u) order E(u) 2 order N N b E(p) order E(p) 2 order TABLE I: The grid reinement analysis or the rotational low problem with µ =.2, = x/4, at t = Y Y.5 1 (a) 1 Velocity ield at t = 1.5 X X (b) Fig. 1: Velocity ield at time t = 1 with a grid, µ =.2, = x/4. The immersed boundary rotates with angular velocity ω = 2. Fig. 1(a) Plot o the x component o velocity ield. Fig. 1(b) Plot o velocity vector ield. V. NUMERICAL RESULTS In this section we present the numerical results or three problems which involve immersed boundaries. A. Rotational low In this problem, the interace is a circle with radius r =.3 embedded in square domain [ 1, 1] [ 1, 1]. We prescribe the interace to rotate with angular velocity ω = 2. We set µ =.2 and consider the solution when t = 1. The velocity ield is shown in Figure 1. We carried out a grid reinement analysis, using a reerence grid o , to determine the order o convergence o the algorithm. The results in Table I show that the velocity is second order accurate and pressure is nearly second order accurate. B. Flow past a circular cylinder In this example, we simulate an unsteady low past a circular cylinder immersed in a rectangular domain Ω = [, 3] [, 1.5]. 1 The cylinder has a diameter d =.1 and its center is located at (1.6,.75). The luid density is ρ = 1. and the reestream velocity is set to unity, U = 1. The viscosity is determined by the Reynolds number, Re d. Simulations have been perormed at Re d = 2, 4, 8, 1, 2 and 3 on computational mesh. We use 4 control points to represent the circular cylinder. At the inlow boundary we speciy the velocity corresponding to the reestream velocity, and a homogeneous Neumann boundary condition is applied at the top, bottom and exit boundaries. The pressure is set to zero at the exit boundary. The pressure ield plots are shown in Figure 2. The results appear to be in good agreement with the other numerical simulations and experimental results. It is important to note that the matrix A, or a closed immersed boundary, is singular. This happens because the pressure inside the close boundary is not uniquely determined. We choose the pressure inside the cylinder such that there is no jump in pressure at one o the control points, i.e., the normal orce at that point is set to zero. Thereore, we can eliminate one column and row o the matrix A corresponding to that control point, thus making the problem solvable. C. Flow past a lat plate In this example, we simulate an unsteady low past a lat plate immersed in a rectangular domain Ω = [, 3] [.75,.75]. The lat plate whose length is L =.1 is oriented in the crosslow direction and located at x = 1.4. Simulations have been perormed at Re L = 2, 5, 1, 1 and 5 on a computational mesh. We use 8 control points to represent the lat plate. The same boundary conditions as or the low past a cylinder problem are applied in this problem. The pressure ield plots are shown in Figure 3. VI. CONCLUSION We have presented a ormally second order accurate immersed interace method or the solution o the incompressible Navier- Stokes equations in irregular domain. The implementation has been tested with three examples involving rotational low and unsteady lows over a circular cylinder and over a lat plate. Numerical experiments have shown that our method can handle problems with rigid boundaries. We plan to incorporate the current approach with our earlier works [16] or problems with deormable interaces to deal with deormable boundaries in irregular domain problems and contact problems.

5 (a) (b) (c) (d) (e) () Fig. 2: Pressure ield at t = 1, = x/4. (a) Re = 2, (b) Re = 4, (c) Re = 8, (d) Re = 1, (e) Re = 2, () Re = (a) (b) (c) (d) (e) Fig. 3: Pressure ield at t = 1, = x/4. (a) Re = 2, (b) Re = 5, (c) Re = 1, (d) Re = 1, (e) Re = 5.

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