An improved fluid-solid coupling method for DNS of particulate flow on a fixed mesh Markus Uhlmann Departamento de Combustibles F´osiles, CIEMAT Avenida Complutense 22, 28040 Madrid, Spain
[email protected]
1
Introduction
Our research is motivated by the current incomplete understanding of the complex phenomena occurring in particulate flow systems such as fluidized bed reactors. By analogy with previous achievements in single-phase turbulence it can be expected that future numerical experiments will provide valuable data in this field. However, the task of simulating a large number of moving phase interfaces has long been considered prohibitive. Only recently have advances in hardware and algorithms promoted a beginning of direct numerical simulations of turbulent flow including the resolution of the fluid-solid interfaces (e.g. Kajishima et al., 2001). Indeed much interest has been devoted to numerical methods which avoid adapting the mesh to the time-dependent position of the particles. In the following we will focus upon the class of so-called fictitious domain methods where the same set of equations is solved in both the fluid and solid domains. In this framework rigid-body motion in the solid domains is enforced by means which differ from author to author, e.g. Lagrangian multipliers (Glowinski et al., 2001), an artificial source term (H¨ofler and Schwarzer, 2000) or a posteriori overwriting of the solution (Kajishima and Takiguchi, 2002). We have chosen to combine elements from the immersed-boundary method of Peskin (2002) and the direct formulation of the artificial force term introduced by Fadlun et al. (2000). The advantage of this approach is that the variation of the hydrodynamic forces acting upon the particle remains smooth while the particle is in arbitrary motion with respect to the fixed grid. Furthermore, the present method does not suffer from the strong limitations of the time step experienced when a feed-back mechanism is used to formulate the force term. The objective of this paper is a further validation and analysis of the method proposed by Uhlmann (2004a). For this purpose we first consider Taylor-Green flow in an immersed circular region. As a second test case we discuss the simulation of a freely-rotating cylinder placed in Couette flow. In § 4 we conclude the paper by summarizing our findings.
1
2
Numerical method
For clarity we will first present the concept of the present method in the framework of a single-step time discretization. The final algorithm will be given in equation (6) below. For this purpose, let us write the momentum equation in the following form un+1 − un = rhsn+1/2 + f n+1/2 , ∆t
(1)
where rhsn+1/2 regroups all usual forces (convective, pressure-related, viscous) and f n+1/2 is the fluid-solid coupling term, both evaluated at some intermediate time level. Since the work of Fadlun et al. (2000) it is common to express the additional force by simply rewriting the above equation as f n+1/2 =
u(d) − un − rhsn+1/2 ∆t
(2)
where u(d) is the desired velocity at any point where forcing is to be applied (i.e. at a point inside a solid body). Formula (2) is characteristic for direct forcing methods. Problems arise from the fact that in general the solid-fluid interface does not coincide with the Eulerian grid lines, meaning that interpolation needs to be performed in order to obtain an adequate representation of the interface. In Uhlmann (2004a) the definition of the force term was instead formulated at Lagrangian positions attached to the surface of the particles, viz. U(d) − Un − RHSn+1/2 , (3) Fn+1/2 = ∆t where upper-case letters indicate quantities evaluated at Lagrangian coordinates. Obviously, the velocity in the particle domain S is simply given by the solid-body motion, U(d) (X) = uc + ω c × (X − xc )
X∈S,
(4)
as a function of the translational and rotational velocities of the particle, uc , ω c , and its center coordinates, xc . The final element of the method of Uhlmann (2004a) is the transfer of the velocity (and r.h.s. forces) from Eulerian to Lagrangian positions as well as the inverse transfer of the forcing term to the Eulerian grid positions. For this purpose we define a Cartesian grid xijk with uniform mesh width h in all three directions. Furthermore, we distribute so-called discrete Lagrangian force points Xl (with 1 ≤ l ≤ NL ) evenly on the particle surface. Using the regularized delta function formalism of Peskin (2002), the transfer can be written as: X U(Xl ) = u(xijk ) δh (xijk − Xl ) h3 , (5a) ijk
f (xijk ) =
X
F(Xl ) δh (xijk − Xl ) ∆Vl ,
l
2
(5b)
where ∆Vl designates the forcing volume assigned to the lth force point. We use the particular function δh given in Roma et al. (1999) which has the properties of continuous differentiability, second order accuracy, support of three grid nodes in each direction and consistency with basic properties of the continuous delta function. It should be underlined that the force points are distributed on the interface between fluid and solid, Xl ∈ ∂S, and not throughout the whole solid domain S. The reason for this is efficiency: the particle-related work scales as (D/h)2 instead of (D/h)3 . In § 3.2.2 we will return to this point when we compare results obtained with both variants. The above method has been implemented in a staggered finite-difference context, involving central, second-order accurate spatial operators, an implicit treatment of the viscous terms and a three-step Runge-Kutta procedure for the non-linear part. Continuity in the entire domain is enforced by means of a projection method. For completeness, the full semi-discrete equations for each Runge-Kutta sub-step (indicated by superscript k ) are given in the following (superscripts (m) refer to a particle in the range 1 ≤ m ≤ Np ): ˜ = uk−1 + ∆t 2αk ν∇2 uk−1 − 2αk ∇pk−1 u k−1 k−2 −γk [(u · ∇)u] − ζk [(u · ∇)u] X (m) ˜β (X(m) ) = U u˜β (xijk ) δh (xijk − Xl ) h3 ∀ l; m; 1 ≤ β ≤ 3 l
(6a) (6b)
ijk
(m)
(m)
F(Xl
) =
fβ (xijk ) =
U(d) (Xl Np NL X X
˜ (m) ) ) − U(X l ∆t (m)
Fβ (Xl
∀ l; m (m)
) δh (xijk − Xl
) ∆Vl
(m)
(6c) ∀ i, j, k
m=1 l=1
1≤β≤3 ∇2 u ∗ −
˜ u 1 u = − + f k + ∇2 uk−1 αk ν∆t ναk ∆t ∇ · u∗ ∇2 φ k = , 2αk ∆t uk = u∗ − 2αk ∆t∇φk , pk = pk−1 + φk − αk ∆t ν∇2 φk ∗
(6d) (6e) (6f) (6g) (6h)
where the set of coefficients αk , γk , ζk (1 ≤ k ≤ 3) is given in Rai and Moin (1991). The intermediate variable φ is the so-called “pseudo-pressure”, u∗ the predicted velocity field; both are discarded after each step. The particle motion is determined by the Runge-Kutta-discretized Newton equations for rigid-body motion, which are weakly coupled to the fluid equations. In the simulations discussed below direct particle interactions (collisions) are not considered.
3
3
Results
Our method has been subjected to a wide array of validation cases: (i) Taylor-Green flow in an immersed region; (ii) uniform flow around a stationary cylinder; (iii) uniform flow around an oscillating cylinder; (iv) a single sedimenting circular disc; (v) a freelyrotating cylinder in Couette flow; (vi) drafting-kissing-tumbling of two circular discs; (vii) pure wake interaction of two circular discs; (viii) sedimentation of a single spherical particle; (ix) pure wake-interaction of two spherical particles; (x) drafting-kissing-tumbling of two spherical particles. Most of these simulations have been documented in (Uhlmann, 2004a,b,c; Uhlmann and Pinelli, 2004). Here we will concentrate on cases (i) and (v). The former case is instructive because it demonstrates the spatial smoothness of the present method. The latter case allows us to verify the correct implementation of the angular particle motion and helps to elucidate the effectiveness of the forcing scheme in the framework of a fractional step method.
3.1
Taylor-Green flow in an immersed circular region
We consider the case of an array of planar decaying vortices with the following analytical solution : 2
2
u(x, y, t) = sin(kx x) cos(ky y)e−(kx +ky )νt kx 2 2 v(x, y, t) = − sin(ky y) cos(kx x)e−(kx +ky )νt ky 1 kx2 2 2 2 2 p(x, y, t) = cos (ky y) 2 − sin (kx x) e−2(kx +ky )νt 2 ky
(7a) (7b) (7c)
where kx =ky =π has been chosen. This flow is simulated in an immersed circular domain with radius unity, centered at the origin of the rectangular computational domain defined by Ω = [−1.5, 1.5] × [−1.5, 1.5]. At the boundary of the computational domain Γ we can apply any type of consistent boundary condition (here: a Dirichlet condition for velocity and a homogeneous Neumann condition for pseudo-pressure was chosen). The exact velocity values from (7) define the “desired” velocity U(d) (Xl ) on the circumference of the immersed circle. Therefore, the present case features one-way coupling and a smooth analytical solution across the immersed boundary, allowing for a strict measurement of the numerical error. In the following, the viscosity is set to ν=0.2 and the equations are advanced for 0 ≤ t ≤ 0.3 using a time step of ∆t=0.001. Figure 1 shows the error of the final velocity as a function of the mesh size h. The error was computed as a maximum over all grid nodes which are located inside the embedded domain. It can be seen that the error increases with the square of the mesh size, which confirms the expected second-order accuracy of the interpolation with the regularized delta function in the case of smooth fields. In order to verify if the present results are sensitive to the position of the immersed boundary relative to the fixed grid, we repeat the previous simulation at a fixed resolution 4
−3
x 10
2.4
−2
max. error
max. error
10
−3
10
2.2
−4
10
−2
2
−1
10
10
0
0.25
h
0.5
0.75
1
xc /h
Figure 1: Maximum error of the velocity field in the case of Taylor-Green vortices. Left graph: The error is shown as a function of the mesh width h without embedded boundary (◦) and with circular embedded boundary (4); the dashed line is proportional to h2 . Right graph: The error is shown as a function of the horizontal position of the embedded circle, for fixed h = 0.05. (h = 0.05) and for different positions of the circle xc , shifted horizontally by fractions of the mesh-width with respect to the origin. The result of this experiment is included in figure 1. The error varies by less than 10% and, more importantly, it exhibits a smooth variation as a function of the shift. This behavior is highly desirable in view of the application to the simulation of the motion of freely-moving particles. It should be mentioned that grid-related interpolation schemes of the same order of accuracy as our present method were found to produce strongly oscillating solutions in similar cases (Uhlmann, 2003).
3.2
A freely-rotating cylinder in Couette flow
In this case a cylinder with diameter D is located in the center of a plane channel whose walls are moving at constant speed in opposite directions (cf. figure 2). The origin of the coordinate system is located on the centerline and the channel walls are spaced H apart. In the absence of the cylinder and for sufficiently low Reynolds numbers the velocity field is given by u = (0, Gy), where G is the shear rate, i.e. the walls are moving at the velocity ±GH/2. The cylinder is translationally fixed but it can rotate freely. The characteristic Reynolds number is based upon the shear rate and the cylinder diameter, ReG = GD 2 /ν. A second non-dimensional parameter is the confinement ratio H/D. Finally, the density ratio ρp /ρf between cylinder and fluid affects the transient behavior. This configuration has been studied numerically by Ding and Aidun (2000) who used a Lattice-Boltzmann method, and experimentally by Zettner and Yoda (2001). In both references the cylinder was neutrally buoyant (ρp /ρf = 1). At steady state, however, the density ratio drops out of the system of equations. We use a uniform grid of 640 × 320 cells (2224 × 256 for the highest Reynolds number 5
uw ...........
y ... ....
x ...........
... ....... ... ... . ... . ... . ... . ... . ... . ... . ... . ... . .. ........ ...
H .................. ....... .... ..... ... ....... ... ... .... ... ... ... ... ... .. ... ... ... . ..... . ........................
... ..... ... . ... . ......... ...
D
..........
uw Figure 2: Schematic of the configuration for a freely-rotating cylinder in Couette flow. 0
increment
10
−5
10
−10
10
−15
10
0
1
2
tG Figure 3: Step-to-step increment of the velocity field ( ) and the rotation rate ( ) as a function of non-dimensional time for a freely-rotating cylinder in Couette flow at ReG =1. In the case of the velocity field the maximum norm is used. value ReG = 100), identical to Ding and Aidun (2000). We focus on a confinement ratio of H/D = 2, which means that the cylinder diameter is resolved by 160 mesh widths (128 at ReG = 100) and 502 (402) Lagrangian force points are distributed on its circumference. The time step was set to ∆t1 =0.001/G such that the maximum CFL number measures approximately 0.2; these results were verified with a reduced time step of ∆t2 =0.0001/G. Additional runs with various values for ∆t were performed for the lowest Reynolds number as discussed in § 3.2.1. The coordinate direction along the channel axis is treated as periodic; the no-slip condition applies at both walls where homogeneous Neumann conditions are imposed upon the pseudo-pressure. The simulation is advanced in time until a steady-state is reached. This was verified by 6
0.5
0.4
θ˙c G 0.3
0.2
0
10
1
10
2
10
ReG Figure 4: Non-dimensional rotation rate θ˙c /G vs. Reynolds number for a freely-rotating cylinder in Couette flow at a confinement ratio of H/D = 2: M, experiments of Zettner and Yoda (2001); +, simulations of Ding and Aidun (2000); ◦, present results, using time step ∆t1 and the same grid as Ding and Aidun (2000). monitoring the maximum norm of the step-to-step increment of the velocity field ||un+1 − un || as well as the increment of the cylinder rotation rate |θ˙cn+1 − θ˙cn |. The convergence of these quantities up to machine accuracy (using 64bit arithmetic) can be verified from figure 3, which is representative of our computations. Figure 4 shows the rotation rate at steady-state as a function of the Reynolds number, (a) ∆t=10−3
(b) ∆t=10−4
Figure 5: Streamlines of the wall-driven shear flow around a freely-rotating cylinder at shear-based Reynolds number ReG = 1 and confinement ratio H/D = 2, computed with two different time steps.
7
ReG 1 1 1 10 100
rotation rate θ˙c 0.4381 −4 ∆t = 10 0.4221 solid domain forcing 0.4219 0.4182 0.2977 comment
max. deviation from rigidity 0.27148 0.03947 0.08275 0.04412 0.01237
Table 1: A freely-rotating cylinder in Couette flow: maximum deviation of the velocity at the fluid-solid interface from a pure rigid-body rotation ||Un+1 (X) − ω c × (X − xc )||/(ωcrc ). The velocity field un+1 at steady-state was transferred to the interface by the interpolation formula given in (5a). The time step was ∆t=10−3 if not otherwise stated.
max. deviation
comparing our values with those from the literature. In the low-Reynolds regime the angular velocity is independent of the Reynolds number and measures θ˙c /G=0.4192 at the present confinement ratio, according to the simulations of Ding and Aidun (2000). It should be noted that theory predicts a value of θ˙c /G=0.5 for vanishing Reynolds numbers and infinite confinement ratio. For values of the Reynolds number beyond 10 the rotation rate decreases significantly. From the figure it can be seen that our results at ReG =10 and ReG =100 match the data of Ding and Aidun (2000) extremely well. The respective errors are below 0.5%, 2%, where it should be noted that the reference value in the latter case was obtained by digitalization of the graphs given by Ding and Aidun (2000) and subsequent linear interpolation (our values are given in table 1 for future reference). At the lowest Reynolds number ReG =1 our method yields an over-prediction of approximately 6% with the larger time step; for the smaller time step ∆t2 the prediction is within 0.7% of the reference data. At first glance it seems curious that a dependence of the steady-state upon
0
10
−1
10
−2
10
−3
10
−5
10
−4
−3
10
10
∆t Figure 6: The maximum deviation of the interface velocity from a rigid-body rotation (ReG =1, cf. table 1) vs. the time step. The dashed reference line indicates a slope of ∆t. 8
the time step is observed and this issue shall be discussed in more detail in § 3.2.1. 3.2.1
Efficiency of the forcing scheme
In general, the updated velocity field un+1 does not exactly reflect the no-slip condition at the fluid-solid interface. Three mechanisms contribute to this effect: 1. Spatial interpolation. The transfer between Eulerian and Lagrangian representations introduces a spatial error of O(∆x) in the case of non-smooth fields (Leveque and Li, 1994). However, this error is minimized in the present case since a very high spatial resolution of the immersed cylinder was chosen. 2. Explicit/implicit formulation. The force term is determined from a fully explicit discretization of the r.h.s. of the momentum equation (6a). However, since in the prediction step (6e) a semi-implicit discretization of the viscous term is used, the intermediate velocity field u∗ does not correspond exactly to the “desired” velocity at the interface—irrespective of the spatial interpolation. 3. Projection. The predicted field u∗ is in general not divergence-free even when un+1 is converged. Hence, it will be corrected during the projection step (6g). This is the reason for the well-known slip-error of order ∆t, generated by the standard fractionalstep method at solid boundaries (e.g. Strikwerda and Lee, 1999). Here, it also affects the locations where the artificial forcing term is finite since it is not guaranteed a priori that ∇ · f vanishes. 1 Figure 5 shows the streamlines obtained at the lowest Reynolds number using the two different time steps. When using the larger step ∆t1 the streamlines visibly cross the cylinder surface which means that impermeability is not imposed effectively. For the reduced step ∆t2 no such defect is noticeable. A more quantitative measure of the efficiency of the forcing scheme is given in table 1 where the maximum deviation from rigid-body motion has been computed at the fluid-solid interface ∂S. For this purpose the velocity at steadystate was interpolated to the interface locations by use of relation (5a), subtracted from the target value ω c × (X − xc ) and normalized by ωc rc . At ReG =1 the error is reduced nearly sevenfold when reducing the time step by a factor of 10, yielding a deviation of approximately 4%. From a logarithmic plot of a series of runs at this Reynolds number (cf. figure 6) it can be deduced that the error scales as O(∆t) asymptotically, as can be expected for the present fractional step method. At higher Reynolds numbers the deviation from rigidity at the interface is much smaller (cf. table 1): roughly 4% and 1% at ReG =10 and ReG =100, respectively, without lowering the time step (∆t1 ). The streamlines plotted in figure 7 confirm the physically correct representation of the rotating solid interface. In particular, the pattern at the highest 1
This observation indicates a possible remedy for the effect: if the force term were formulated in a divergence-free basis by design then it would not cause any additional correction. A modification along this line has not been attempted so far.
9
(a) ReG =10
(b) ReG =100
Figure 7: Streamlines of the wall-driven shear flow around a freely-rotating cylinder at shear-based Reynolds numbers ReG =10 (left graph) and ReG =100 (right) and confinement ratio H/D = 2, computed with time step ∆t=10−3 .
Figure 8: Streamlines of the wall-driven shear flow around a freely-rotating cylinder at shear-based Reynolds numbers ReG =1, computed with time step ∆t=10−3 and collocating Lagrangian force points throughout the solid domain S. value of the Reynolds number is very similar to the data given by Ding and Aidun (2000) at ReG =76.8 (cf. their figure 3). Why is the efficiency of the forcing higher at higher Reynolds numbers? Concerning the second contribution in the above list, it is obvious that its magnitude decreases with the viscosity, tending to zero in the inviscid limit. The deviation due to the third contribution decreases for higher Reynolds numbers since the magnitude of the forcing term itself diminishes with viscosity, at least in the regime of low Reynolds numbers. An evaluation of the maximum of f shows a sevenfold reduction between ReG =1 and ReG =10; at the same time the divergence term ∇ · f drops by a factor of seven. This explains why the choice of the time step has a smaller influence upon the results at these values of the Reynolds number.
10
(a) interface forcing
(b) solid domain forcing
Figure 9: Isocontours of the pressure field of the wall-driven shear flow around a freelyrotating cylinder at shear-based Reynolds number ReG = 1. The pressure fields are normalized by their respective maximum value and isocontours are plotted at values −1(.1)+1. 3.2.2
Comparison between interface forcing and solid-domain forcing
Here we will consider results obtained when Lagrangian force points are located not only on the fluid-solid interface ∂S but also throughout the solid domain S (cf. Uhlmann, 2004b). Hence, even at the inside of the particle the rigid-body velocity is imposed. From table 1 it can be seen that—at the lowest Reynolds number value—this modification has the effect of substantially improving the prediction of the rotation-rate (0.6% error) and that it yields a reasonable deviation from rigidity at the interface of approximately 8% at the larger time step. In fact, the deviation from rigid-body motion at any grid-node inside the immersed cylinder is below 1.5%. This latter error is smaller than the interface error since no spatial interpolation is necessary for its evaluation. Figure 8 shows the corresponding streamlines which exhibit a pattern nearly identical to the result obtained with interface forcing and using the reduced time step (cf. figure 5b). The pressure contours for both types of force-point collocations are visualized in figure 9. In both cases a vanishing gradient normal to the immersed boundary is obtained on the fluid side of the interface, yielding a very similar pattern of contours throughout the fluid region. In the solid region, however, the pressure fields are fundamentally different. With interface forcing an approximately constant pressure is generated inside the cylinder, i.e. a strong jump across the interface is observed. Domain forcing, on the other hand, leads to a continuous pressure across the interface. One consequence is that the forcing term— although of similar magnitude—has a much smoother spatial distribution in the latter case and thereby its divergence is greatly reduced. This in turn means that the deviation due to the projection step (cf. item 3 in § 3.2.1) decreases. It would be interesting to compare the transient behavior of the two variants. Unfortunately, no reference data is available in the present case. For a similar comparison in the case of the sedimentation of a single circular particle the reader is referred to the previous study of Uhlmann (2004b). 11
4
Conclusions
We have discussed different aspects of a recently proposed fictitious domain method which uses the regularized delta function of Peskin and co-workers (Peskin, 2002; Roma et al., 1999) for the association between arbitrary Lagrangian and discrete Eulerian positions. In the present approach the fluid-solid coupling force is formulated explicitly in the context of a fractional step method. The scheme has been implemented in a semi-implicit secondorder finite-difference solver. Considering the case of Taylor-Green flow in an immersed circular domain allowed us to demonstrate the second-order accuracy of the overall scheme in the case of smooth fields. Furthermore, it was shown that the error does not depend critically upon the location of the interface with respect to the fixed grid, a property which is vital once the interface is set in motion, and which is not guaranteed when using standard second-order interpolation formulas. When the solution has discontinuous derivatives, however, Leveque and Li (1994) have shown for simple elliptic equations that second-order spatial accuracy in more than one space dimension cannot be achieved at all grid-points by methods based upon the regularized-delta-function approach. We have analyzed the case of a freely-rotating circular cylinder in Couette flow in detail. Our results for the steady-state match those of the reference data within few percent error. Moreover, the efficiency of the present forcing method could be gauged by measuring the deviation from rigid-body motion at the fluid-solid interface. It was shown that there is a time-step dependence of the results at steady-state, mainly due to the effect of the projection step. This effect significantly decreases with the value of the Reynolds number in the investigated interval. Finally, we have compared results of the interface-forcing strategy with those obtained when forcing the whole solid domain. In the latter case a reduced deviation from rigid-body motion is obtained at the interface at low Reynolds numbers; this is due to the smoother spatial distribution of the force term and its smaller divergence. On the other hand it should be taken into consideration that the computational cost of forcing the entire solid domain is larger: the particle-related work then scales as (D/h)3 instead of (D/h)2 . Current applications of the present method include the simulation of the continuous sedimentation of O(1000) spherical particles in a tri-periodic domain and the simulation of dilute particulate flow in a plane channel configuration in the fully turbulent regime.
Acknowledgements This work was supported by the Spanish Ministry of Education and Science under the Ram´on y Cajal program (contract DPI-2002-040550-C07-04) and through grant DPI-20021314-C07-04.
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