An added mass partitioned algorithm for rigid bodies and incompressible flows Jeff Banks Rensselaer Polytechnic Institute Overset Grid Symposium Mukilteo, WA October 19, 216
Collaborators Bill Henshaw, Don Schwendeman, Qi Tang Department of Mathematical Sciences Rensselaer Polytechnic Institute Support Department of Energy Office of Advanced Scientific Computing Research Applied Mathematics Program NNSA Rensselaer Polytechnic Institute
In recent work we have focused on large deformation and/or displacement FSI using partitioned solvers Component solvers remain independent can use existing solvers no need to solve (or precondition) a coupled implicit system Can naturally take advantage of disparate time scales e.g. mixing implicit and explicit integration High levels of algorithmic concurrency maps well to modern and emerging computers
In recent work we have focused on large deformation and/or displacement FSI using partitioned solvers fluid solver interface solid solver Component solvers remain independent can use existing solvers no need to solve (or precondition) a coupled implicit system Can naturally take advantage of disparate time scales e.g. mixing implicit and explicit integration High levels of algorithmic concurrency maps well to modern and emerging computers
Traditional partitioned schemes have suffered from added-mass instabilities for solids which are sufficiently light when compared to the fluid Traditional partitioned FSI algorithms (Cirak, et. al. 27, Bungartz and Schafer 26) 1. advance fluid (using interface velocity/displacement from the solid) 2. advance solid (apply fluid forces to the solid) 3. possibly iterate with under-relaxation to convergence t (n) fluid force displ./vel. structure t (n+1) diagram from Keyes et. al. 212 underrelaxation interface quasi Newton Some analysis of added-mass instabilities can be found in the literature, for example Causin, Grebeau, and Nobile, 25 (stability with relaxation) Gretarsson, Kwatra, and Fedkiw 211 (semi-monolithic formulations)
The origin of added-mass instabilities is that the effect of displaced fluid is not appropriately accounted for in the numerical algorithms in a vacuum force Body simply moves according to Newton s laws of motion
The origin of added-mass instabilities is that the effect of displaced fluid is not appropriately accounted for in the numerical algorithms in a vacuum force Body simply moves according to Newton s laws of motion
The origin of added-mass instabilities is that the effect of displaced fluid is not appropriately accounted for in the numerical algorithms in a vacuum in a fluid force force Body simply moves according to Newton s laws of motion
The origin of added-mass instabilities is that the effect of displaced fluid is not appropriately accounted for in the numerical algorithms in a vacuum in a fluid fluid must be displaced force force Body simply moves according to Newton s laws of motion
The origin of added-mass instabilities is that the effect of displaced fluid is not appropriately accounted for in the numerical algorithms in a vacuum in a fluid force force fluid must be entrained Body simply moves according to Newton s laws of motion
The origin of added-mass instabilities is that the effect of displaced fluid is not appropriately accounted for in the numerical algorithms in a vacuum in a fluid fluid contributing to added mass force force Body simply moves according to Newton s laws of motion Body must displace and entrain fluid to move and therefore appears more massive than in vacuum... the so called added mass
As a concrete motivating example consider a rising rigid body in counterflow Incompressible Navier-Stokes Light rigid body flow
This case has both strong added-mass and added-damping effects Added Mass Added Damping M a n p t m b G I z F Added mass relates to increased apparent mass owing to fluid displacement i.e. the local geometry occupied by solid changes Added damping relates to increased apparent inertia owing to viscous fluid drag i.e. the local geometry remains fixed
To understand these effects in isolation we derive extremely simple models by localizing and linearizing the problem near the interface y-translations relate to added-mass effects x-translations relate to added-damping effects
The resulting model is used to motivate our new AMP algorithms and discuss the performance of traditional partitioned (TP) schemes y = H fluid: y = interface: b rigid body: b y = H x = x = L Fluid: Rigid body: m b a u = m b a v = @v + rp = µ @t v, x 2, r v =, x 2, Z L Z L µ @u @y (x,,t) dx + g u(t), p(x,,t) dx + g v (t), Interface: v(x,,t)=v b (t), x 2 [,L], 2
An added-mass model problem is derived by considering vertical motions y = H y = y = H x = fluid: interface: b rigid body: b x = L 8 @v @t + @p @y =, >< @v @y =, dv b m b dt = >: v(,t)=v b, Z L pdx, p(h, t) =p H (t), y 2 (,H), y 2 (,H), The AMP scheme matches the vertical accelerations at the interface @v @t y= = dv b dt = a And applies a generalized Robin condition to the fluid pressure equation a + @p @y = m b a = Z L pdx
The resulting AMP scheme is stable for any finite mass, while the traditional scheme suffers y = H y = y = H x = fluid: interface: b rigid body: b x = L 8 @v @t + @p @y =, >< @v @y =, dv b m b dt = >: v(,t)=v b, Z L pdx, p(h, t) =p H (t), y 2 (,H), y 2 (,H), The added mass for this problem is easily identified as M a = LH Thm: The 2nd order accurate AMP scheme for the added-mass model problem is stable provided m b + M a is bounded away from zero. Thm: The 2nd order traditional partitioned scheme is stable if and only if m b >M a
An added-damping model problem is derived by considering horizontal motions y = H y = fluid: interface: b rigid body: b 8 >< @u @t = u µ@2 @y 2, Z L du b m b dt = µ >: u(,t)=u b (t), @u (,t) dx, @y u(h, t) =u H (t), y 2 (,H), y = H x = x = L The AMP scheme uses the exact solution to form the discrete approximation µ Z L @u (,t) dx @y Du b D µ Z L 1 e y = y p t/2 here is a ratio of viscous length scales, and D is an added-damping coefficient (in 3D these become added-damping tensors)
The AMP scheme with extra velocity projection is stable even for massless bodies y=h fluid: interface: b y= AMP VC, A=4.5, β=.25 1 n.5 Re(ξ ) 1 Re(A ) (scaled) rigid body: b j y= H x=.5 8 @u @2u > > = µ 2, y 2 (, H), > > @y > < @t Z L dub @u > mb =µ (, t) dx, > > AMP VC, A=.84+.62i, β=.31 AMP VC, A=.81+.65i, β= 6. dt > @y > 1 : u(, t) = ub (t),iv u(h, t) = uh (t), II.5 x =L.5 Figure 3: The geometry for the rectangular geometry FSI.5 model problems. I Re(An) (scaled) 1 5 1 j, n 15 1 2.5 Re(ξj) 5 j, n 1 Re(An) (scaled) 1 Re(ξj) ic equation for the fluid pressure can be derived by taking the divergence of the momentum 1) and using the continuity equation in (2) to give p=, xstable 2. region (9) ty-pressure form of the Stokes equations, the momentum equation for the velocity in (1) is used place equation for the pressure in (9) instead of the divergence equation in (2). II n consistency with the original velocity-divergence form of the equations, zero divergence is an additional boundary condition, i.e. r v = for x 2 @, see [23] for additional details. In n to follow we will use a fractional-step method based on the velocity-pressure form of the fluid define partitioned FSI algorithms. In Section 3 a traditional partitioned scheme, referred to as IV added mass partitioned scheme algorithm, is described. Then in Section 4 we present our new e referred to as the AMP-RB algorithm. The important di erence in the two approaches lies in of the fractional-step method for the fluid with the equations governing the motion of the body. I Stable region onal Partitioned/Rigid-Body algorithm TP-RB n our discussion of partitioned algorithms for the model FSI problem by considering a traditional 5 j, n 1
The AMP-RB scheme is implemented in Overture and is found to be stable against both added-mass and added damping instabilities without iteration b =.1 p t =2..223.242 p t =4..141.444 p t =1..26.719 G
.157 a G4.156.5.155 b.66 ab G8 ab G16 ab G32 By comparison, the traditional partitioned scheme requires ~85 under-relaxed iterations to provide comparable results.154.6.65.7.75.8.68.85 t.7 ab G4 a G8.72 ab G16.74 b.5 a G32 b.1 2 b =.1 4 6 t ab G4 ab G8 ab G16 a G32 b 8.76 12 4.4 1 4.5 4.6 4.7 4.8 4.9 5 t Figure 11: Light rising body. Left: vertical acceleration of the light rising body. Right: zoom near t = 4.75 the perturbations to the accelerations caused by the changes in overlapping grid interpolation points as moves. the surface integrals defining the added-damping tensors (57)-(6), using the discrete geometry defining the rounded rectangle. For grid G (4), these computed added-damping coefficients for the rounded rectangle were found to be µ µ µ vv vv!! D11 1.917, D22.917, D22.379, n n n p t = 4. p t are t = 2. = 1. withp other entries in the tensors being approximately zero. These above entries nearly equal to the actual coefficients for a square-cornered solid rectangle (??) of.223.242 vv D11 = 2w µ µ =2, n n.141 vv D22 = 2h.444 µ µ =, n n.26!! D22 = wh(w + h)/2.719 µ 3 µ µ = =.375. n 8 n n Risplay ingabody, b =.1 For this problem, added-damping e ects can critical ρrole, especially on coarser grids p when the time.7 step is larger. Recall that added-damping is proportional to µ t/ n, where n = t/2 and thus p yb (4) added-damping e ects increase for larger values of t. Figure 13 shows that the simulation on grid Gfb.6 vb e ects when the added-damping coefficient d is chosen too small ( d =.5 is unstable due to added-damping in this case). The instability appears primarily in the rotation of the body, the time history of the angular ab acceleration,! 3, shows.5 a large high-frequency oscillation. This instability seems to become saturated due to yb TP SI counter-acting pressure forces (i.e. added-mass e ects). vb TP SI Figure 12 compares results from the AMP-RB scheme versus the TP-SI on grid G (4). The TP-SI scheme.4 required on average 85 sub-iteration per step with a relaxation parameter of! =.25. The results are ab TP SI nearly indistinguishable, indicating that the AMP scheme can achieve essentially the same solution as the.3 without sub-iterations. traditional scheme but Figure 9: Rising body, streamlines and contours of the pressure for b =.1 at times t = 2, t = 4 and t = 1 computed using grid Grid G (16)..2.1.7.6 Rising body, ρ b =.1 yb G4 yb G8 yb G16.5 Ris ing body, ρ b =.1.18.16.1 2 Ris ing body, ρ b =.1.4 Figure 8: yb G4 Ris ing body, ρ b =.1 6 t.14 yb G32.4 4 8.166.1655 1.165.1645.12.164 Figure 12: A comparison of the AMP-RB scheme versus the TP-SI for a light body. Results for grid G (4).1t = Rising this body: difficult Coarse composite grid Gthe, attp-si times 2, t = 4 and t = 1. Aon gridaverage, is located on top problem, scheme required, 85thesub-iterations per step. The results ar.399.398 yb G8 yb G16 yb G32.1635.163.1625 v G4
A more challenging case is that of a light cylinder rising in counterflow The AMP-RB scheme is again stable without any iteration Traditional partitioned scheme needs ~2 sub iterations to stabilize b =.1 y 1 2 1 2 x 5 4 5 4
A more challenging case is that of a light cylinder rising in counterflow b =.1 flow
Extensions to multiple bodies presents no particular challenges, and the scheme is robust even in the presence of both light and heavy bodies
Extensions to multiple bodies presents no particular challenges, and the scheme is robust even in the presence of both light and heavy bodies
Summary Careful analysis of simplified model problems motivates our stable and 2nd order accurate AMP-RB scheme for incompressible flows Stability analysis in simple geometries shows excellent stability properties even in the uniterated form Implementation within Overture illustrates the utility of the approach for both light and heavy bodies Future Work implement in 3D investigate the alternate added-mass potential formulation New FSI regimes (incompressible/incompressible, compressible/beams)