Flow Past Tandem Circular Cylinders at High Reynolds numbers … · 2017-12-22 · Flow Past Tandem Circular Cylinders at High Reynolds numbers using Overset Grids in OpenFOAM Dominic

Flow Past Tandem Circular Cylinders at High

Reynolds numbers using Overset Grids in OpenFOAM

Dominic Chandar ∗and Vinh-Tan Nguyen †

Fluid Dynamics, Institute of High Performance Computing, Singapore 138632

Harish Gopalan ‡

Mechanical Engineering, Union College, 807 Union Street, Schenectady, NY 12308

Rajeev Jaiman §

Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576

This study investigates the application of overset grids for simulating high Reynoldsnumber flows in OpenFOAM. The problem of tandem cylinders is specifically chosen asan appropriate test model due to (a) a direct relevance to the marine industry and (b)the availability of experimental results conducted at the Basic Aerodynamics ResearchTunnel at LaRC. Usage of overset grids in OpenFOAM has previously been demonstratedonly using SUGGAR++, and this library is not easily available to the general public.This paper thus explores the application an in-house overset grid library (Overset ParallelEngine for Aerodynamics Applications) OPErA in OpenFOAM. For verification purposesof the overset coupling, results are presented for the inline oscillation of two cylinders in lowReynolds numbers Re 40. For the high Reynolds number case of Re 1.66×105, an assesmentof the forces and the flow between the cylinders are presented for static tandem cylinderswith a distance to diameter spacing of (L/D = 1.4). This specific spacing corresponds toa sub-critical spacing (short separation) where the drag on the rear cylinder is negative.For single grid computations, force predictions for (a) a non-linear URANS model and (b)a hybrid model are presented. For overset grid computations, only non-linear URANSresults are presented to verify the implementation of the overset coupling. Comparisonsbetween single grid and overset grid cases for the forces and the flow features are veryencouranging and demonstrate greater potential in using overset grids for moving bodyproblems.

I. Introduction

The flow past circular cylinders in tandem arrangement has been investigated by several researchers dueto its applications in offshore engineering, noise prediction, ventilation, etc. A comprehensive review of

this topic can be found in Zdravkovich1 . According to Zdravkovich1 , the flow regime can be separatedinto short, intermediate and and large depending on the ratio of spacing between the cylinders and cylinderdiameter (L/D). In the shore separation regime (L/D ≤ 2.4), both the cylinders behave like a single bluffbody. Vortex shedding is observed from the downstream cylinder only. In the intermediate separationregime (2.4 ≤ L/D ≤ 4 − 6), the flow is bi-stable and switches between the short and large separationregime depending on the L/D ratio. Intermittent or constant vortex shedding is observed depending on theL/D ratio and the experimental conditions. Recent experiments performed at Basic Aerodynamics Research

∗Scientist, Fluid Dynamics, Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis North, Singa-pore 138632†Scientist, Fluid Dynamics, Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis North, Singa-

pore 138632‡Senior Research Fellow, Dept. of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singa-

pore 117576§Assistant Professor, Dept. of Mechanical Engineering, National University of Singapore, Block EA, 9 Engineering Drive 1,

Singapore 117576

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American Institute of Aeronautics and Astronautics

Tunnel (BART) at LaRC2 showed that the intermittent vortex shedding occurs upto L/D = 3.0375. The bigdifference between two regions is that, the sign of the drag force on the downstream cylinder switches fromnegative to positive when the shedding becomes constant. Finally, in the large separation regime, constantvortex shedding occurs from each of the cylinders and the wake interference due to the spacing does notexist anymore. It is of tremendous interest to be able to accurately model all the three regimes to developa basic understanding of the flow physics.

Of particular interest in the current investigation is the development of a computational model forinvestigation of the flow interference effects of floatel connected to a turret moored floating production storageand offoading (FPSO) unit. When a floatel is positioned and connected to a turret moored FPSO, the FPSOwill undergo low frequency yaw motion. In order to maintain continuous connection between the FPSO andfloatel, dynamic positioning system (DPS) installed on the floatel needs to be capable enough to follow theFPSO low frequency yaw motion. Due to the proximity effects of the floatel and FPSO, there is significantflow interference and shielding due to wind, wave and currents. These effects will increase the loads on thefloatel. Additionally, the thrusters mounted underneath a floatel will suffer from thruster/hull interactionand the interference due the proximity to the FPSO. Field and wind tunnel measurements for such complexproblems are usually limited in number and the data provided from the experiments are mostly hydrodynamicforces. It becomes very expensive to perform experimental studies for multiple configurations to optimizethe floatel shape and orientation for minimum drag. Hence, measurements have to be supplemented bynumerical simulations. As most of these are high Reynolds number turbulent flows, confidence in the useof the numerical tool has to be developed before simulating the actual problem. This is usually done byvalidating the models for a canonical flow configuration, which can be taken as the simplified setup of theactual problem. Turbulent flow past tandem cylinders is chosen as the canonical flow configuration due tosimilarity of the flow physics to the actual problem and the availability of experimental data for comparison.There are two issues to deal with when solving these kind of problems:

1. Prediction of the forces using standard URANS models are not accurate: Most URANS models arecalibrated using simple canonical attached flows. The predictions of the models for complex flowsare usually less successful. Second, only mean flow data is available from URANS simulations. Theunsteady flow data is essential for many applications. As an alternative to URANS, large eddy sim-ulation (LES) has been receiving a lot of interest in the recent years. LES has been successful inthe simulation of wall boundary flows at low to moderate Reynolds numbers. LES results are usuallyaccurate to within 5-10% of direct numerical simulation data and experimental results. However, thecomputational cost of LES scales as N ≈ Re1.76,3 for wall-bounded flows. This makes it impossible toperform LES of wall-bounded flows at high Reynolds numbers. The high cost of LES modeling and thelimitations of RANS, has led to the development of hybrid RANS-LES methods. The hybrid methodstake advantage of the fact that LES computations are cheaper away from the wall (Nles ∼ Re0.4 foroff-body) and that RANS methods need to be employed only in the near-wall region. Hence, using thehybrid methods, the disadvantages of LES (cost) and RANS (steady) are overcome and the compu-tations are relatively cheaper. It was shown4 that the ratio of the cost of LES to hybrid simulationsscale as Re0.46, providing the ability to simulate high Reynolds numbers flow which were not possiblebefore. A comprehensive review of the application of hybrid methods to complex industrial problemsat high Reynolds numbers can be found in Spalart.5

2. Simulating the free motion of bodies for large displacements poses a restriction on existing dynamicmesh algorithms: In OpenFOAM, the motion of bodies is either coupled with a sliding mesh model,mesh deformation or a more efficient technique known as mesh morphing. However, all mesh defor-mation methods suffer from robustness issues when it comes to large displacements. Cells become tooskewed or the mesh has to be regenerated. Application of overset or overlapping mesh techniques tothese kind of problems is an elegant solution, since each body of interest has its own grid termed asa component grid and a corresponding motion. While component grids move, their motion is alwayscontained within a background undeformed grid, and solutions are interpolated at appropriate pointsin the computational domain. While OpenFOAM has an interface to commericially available tools suchas Suggar++6 and DiRTlib7 termed as foamedOver,8 this application is not available to the generalpublic. With OpenFOAM being a very powerful open-source CFD tool to the engineering community,it would be beneficial to develop an overset grid assembly tool independently, and have it interfacedwith OpenFOAM.

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The aims of the current study are two-fold:

• To simulate the flow past tandem cylinders at a sub-critical spacing of L/D = 1.4 using URANS andhybrid RANS-LES methods on single grids, and

• To assess the applicability of overset grids for these problems.

For URANS computations, the non-linear K − ω SST model based on the constitutive relation proposed byAbe9 (NSST) is used. For the hybrid computations, the non-linear model of Gopalan10 based on the unifiedmodel of Heinz4 is used. Overset grid coupling with OpenFOAM was achieved using the tools in OPERrA.11

The rest of the study is organized as follows. Section 2 discusses the governing equations for incompressibleflows. In section 3, a description of the overset semantics will be provided in detail with an example tounderstand the requirements of the overset grid algorithm. In section 4, the parallel donor search procedureusing the inverse map approach12 will be discussed. In section 5, a verification test to show-case the use ofoverset grids for dynamic problems is presented. Section 6 provides comparative performance of URANS vs.the hybrid model for single grid and comparisons between single and overset grids for the URANS model.Section 7 concludes the story and cites possible future work.

II. Governing Equations

The filtered conservation of mass and momentum equations are given by

∂ui∂xi

= 0 (1)

∂ui∂t

+∂ (uiuj)

∂xj= −1

ρ

∂p

∂xi+ ν

∂2ui∂xj∂xj

− ∂Dij

∂xj(2)

where Dij = uiuj − uiuj is the hybrid stress tensor for the hybrid model and Reynolds stress tensor for theURANS model respectively. It should be noted that there are some assumptions in the above equations.First, filtered and averaged variables are assumed to be equivalent in the near-wall RANS region. Second, thenear-wall RANS region is different from traditional RANS. The kinetic energy is both resolved and modeledin the RANS region. Although, these assumptions seem to violate the basis for the derivation of the RANSmodels, these equations have been successfully applied to a number of complex flows suggesting that theassumptions may hold in a weak sense. The closure of the above set of equations require the specification ofthe stress tensors. For the linear URANS and hybrid models, the stress tensor takes the following form: isgiven by

Dij =2

3kδij − 2νtSij (3)

Here k is the turbulent kinetic energy, νt is the turbulent viscosity, and Sij is the symmetric part of thevelocity gradient tensor. For non-linear URANS models, the stress tensor includes an additional non-linearcomponent and the form of the equation becomes

Dnlij =

2

3kδij − 2νtSij +Nij (4)

where Nij is the non-linear part of the stress tensor. For incompressible flows, the turbulent kinetic energyterm gets absorbed into the pressure term and the Poisson equation is solved for the modified pressurep∗ = p + 2/3k. Hence only the deviatoric part (Dij − 2/3kδij) of the stress tensor is required for theincompressible flow simulations. In the above equations, νt, k and Nij are unknown. The calculation ofthese parameters depend on the choice of the turbulence models. The equations for the models are providedin Gopalan10

III. Overset Grid Semantics

An overset grid is composed of a set of individual and independent component grids gk. Each gk canhave an arbitrary number of parallel partitions pgk which is obtained using the graph partitioning softwareMETIS.13 Consider a set of grids as shown in figs. 1(a)-(c).

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(a) The complete setup showing various grids (b) Field and fringe cells for the near-body grid

(c) Field and fringe cells for the background Grid

Figure 1. Example overset grid for the flow past tandem cylinders

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The figures show a case set-up for the flow past tandem cylinders with three grids. Each grid in thissetup has just one partition for simplicity. The classification of various computational cells in the domainare given below. Each type of cell also has a flag associated with its type - iBlank.

• Field Cells: Cells colored in red indicating that the governing equations for fluid flow are solved.(iBlank = 1)

• Mandatory Fringe Cells: Cells colored in blue which lie near the outer boundary. Mandatory interpo-lation of these cells from neighboring grids are required. A cell that provides data from the overlappinggrid for the purpose of interpolation is termed as a donor and the process of finding donors is calleddonor search. (iBlank = −1)

• Implicit Fringe Cells: Cells also colored in blue which interpolate from neighboring grids. The fringecells are implicit because these are computed automatically based on the overlap strategy (iBlank =−1).

• Hole Cells: Cells colored in green, mainly from a background grid that lie inside a wall boundary of anear-body grid. Field variables are not updated in these cells.(iBlank = 0)

The major goal of the overset grid algorithm is to compute this classification in parallel given a set ofgrids that overlap with each other. Some of the definitions given below will also be used during the oversetgrid assembly:

• Direct Map (DM): This is a map between any arbitrary point in space to an underlying Cartesian grid.Given a point P (x), the map establishes the index co-ordinates of the Cartesian cell that contains thispoint. In one-dimensions, it is illustrated in fig. 2

Figure 2. Illustration of a direct map in one-dimensions

• Inverse Map(IM): This is a map between an arbitrarily oriented Cartesian auxiliary grid and a physicalgrid for a given grid partition. It will be explained later that there is an equivalent auxiliary Cartesiangrid associated with the physical unstructured partitioned grid. Thus, given a cell in a Cartesianauxiliary grid (a query cell), the inverse map should be able to establish a physical cell that is nearto the query cell. However, this physical cell which is associated with the query cell need not be thenearest. Thus the IM can be conveniently expressed as :

CellID_PhysicalGrid = inverseMap[CellID_AuxiliaryGrid]

IV. The Overset Grid Algorithm

The procedure used in this work for computing the overset grid connectivity is based on the work byRoget and Sitaraman,12 but is directed towards making the approach modular so that it can be portedto Open-Source CFD codes such as OpenFOAM. Following Roget and Sitaraman,12 the steps required toestablish an overset grid are given below. There are minor variations to their implementation,12 but theoverall approach is similar.

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1. Generating component grids using the Open-Source Grid Generator GMSH14

2. Partitioning each grid using METIS.13 Each grid will have at least one partition so that the minimumnumber of partitions is equal to the number of grids.

3. Computation of axis-aligned(AABB) bounding boxes and hole boundary identification.

4. Donor search pre-processing

(a) Computation of oriented(OBB) bounding boxes

(b) Intersection Checks between OBBs

(c) Inverse map computation based on an auxiliary Cartesian grid.

5. Donor search between intersecting partitions

6. Collective decision making strategy to classify cells based on the donor search results

7. Interpolation of variables

Step 7 is generally performed within the linear solver and serves as an extra level of communication overheadfor each iteration of the linear solver. When grids move, steps 3-7 are repeated every time step.

IV.A. Grid Generation and Partitioning

GMSH is used as the primary grid generator for OPErA. Although OPErA can read in the grid generated byOpenFOAM using a custom utility, it does not however support polyhedral cells, like those generated usingOpenFOAM’s snappyHexMesh utility. An important procedure to be followed during the grid generationprocess is that, the user has to specify which boundary patches in the grid are mandatory interpolatingpatches. The cells adjacent to the faces in these patches have to interpolate from a neighboring overlappinggrid. In GMSH, this can be achieved by specifying a Physical Type for the boundary patch. e.g.

// i1, i2, i3 are a list of faces in the geometry that are

// mandatory interpolating faces

Physical Surface("OVERSET")={i1,i2,i3..}

This will instruct GMSH to flag all the faces that are contained in the patch named ”OV ERSET”.

IV.B. Axis-Aligned Bounding Box(AABB) Computation and Hole Boundary Identification

After the partitions have been determined, the overset pre-processor routines are called to start computingthe connectivity. The primary step in this process is to establish a set of bounding boxes for the partitionsthat encompass the hole boundaries. For simplicity, axis-aligned bounding boxes are used for this procedure.For each partition in parallel, an AABB is computed only if it has wall boundary nodes. For example, fig.3 shows the AABBs for the partitions that contain wall nodes.

Following this, each partition broadcasts the AABB if its exists to all processors so that a global boundingbox for each hole in the domain can be obtained. The global bounding box for each hole is computed bymerging individual bounding boxes which correspond to the same hole. The global bounding box is againthen broadcasted to all processors. Next, a Cartesian auxiliary grid is established based on the AABB. Thenumber of cells in this auxiliary grid can be a variable and is chosen by the user. As a thumb rule, this sizecould be equal to the size of the background mesh encompassing the hole. A flag termed as the holeTag isinitialized for each auxiliary grid cell. The following algorithm will then flag the holeTag corresponding tothe wall nodes.

For H : All Holes in the Domain

For F : Each Wall Boundary Face

[min max] = getMinMaxOfCoordinates(F)

[min_indx max_indx] = directMap(min,max)

For I : min_indx to max_indx

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Figure 3. Axis-aligned bounding boxes(AABB) for the wall partitions

holeTag[H][I]=isTaggedByWall /* ( = 1 ) */

Done

Done

Done

Using a flood-fill algorithm,12the auxiliary grid cells that are inside the hole, but not on the hole boundaryare flagged. At the end of this step, auxiliary grid cells with a tag of 1 will have a wall boundary face passingthrough it, and cells with a tag 0 will correspond to cells which have all of its nodes inside the wall boundary.

IV.C. Donor search Pre-Processing

IV.C.1. Oriented Bounding Box Computation and Intersection

Similar to the computation of AABB, the pre-processing step involves computing oriented bounding boxes(OBB). An OBB is a tight-fit bounding box that contains the complete grid partition. The OBB is computedfor all partitions, unlike the AABB which is computed only for partitions that contain wall nodes. Examplesof oriented bounding boxes for one of the tandem cylinders for each partition is shown in fig. 4. Thecylinder in this case is partitioned using four processes, out of which partition 1 and 2 contain wall cells.Following the computation of OBBs, intersection checks between the OBBs of different grids are carriedout. For each partition an intersection flag is created and is assigned to true if it intersects with otherpartitions not belonging to the same grid. Given a partition, and a cell within it, a donor for this cell willbe searched only from a partition that it intersects with. Once this has been established, the inverse map(IM) described previously is computed for each auxiliary Cartesian grid. For the donor search to be robust,a query Cartesian cell must associate itself with a nearby physical cell. The reason that we are interested ina query cell which is Cartesian is because, given an arbitrary point P, one can always establish a Cartesiancell that contains this P using the DM in Sec. III. Thus given any arbitrary point in space P, we have thefollowing map:

Given P --> DM[P] --> CellID_AuxiliaryGrid --> IM[CellID_AuxiliaryGrid] --> CellID_PhysicalGrid

The arbitrary point P is usually the cell-center of a cell that needs to find a donor from a grid that it overlapswith, and the CellID PhysicalGrid is nothing but a starting guess for the donor cell.

IV.C.2. The Inverse Map Process

As stated before, an inverse map establishes a relationship between a Cartesian auxiliary grid (computedfrom the OBB of a grid partition) and the actual physical grid partition. The inverse map is established asfollows:

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(a) Partition 0 (b) Partition 1

(c) Partition 2 (d) Partition 3

(e) Complete set of oriented bounding boxes

Figure 4. Oriented bounding boxes(OBB) for the tandem cylinder case

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For i : All Cells In a Given Partition

[min_c, max_c] = minMaxValue(nodes of a cell)

[min_i, max_i] = minMaxIndex(min_c,max_c)

For j : min_i to max_i

iMap[j] = i

Done

Done

i.e. For a given cell and its nodal co-ordinates, we first compute its extrema using minMaxV alue. Using thedirect map strategy, we establish which index of the auxiliary grid it corresponds to using minMaxIndex.Following this, we associate a physical cell i with a Cartesian auxiliary cell j. At the end of this loop, allCartesian auxiliary cells will be associated to some physical cell - vice-versa. It can also be seen that thisprocess can associate multiple physical cells to the same Cartesian auxiliary cell. When that happens, thefinal physical cell will only be stored.

IV.D. Donor Search

Donor search is a process of associating a valid cell on grid gi with a recipient point in grid gk. The recipientpoint in this case would be the cell center of an interpolation cell (mandatory or implicit). As the donorsearch involves considerable use of grid connectivity data on the donor grid, it is more efficient in terms ofmemory, that a donor partitioned grid search for valid receivers from intersecting partitions rather than tohave receiver partitions search for donors across several intersecting partitions. Thus, to begin with, eachgrid partition would act as a donor partition and each partition would have cell data about the intersectingpartitions. The cell data includes the cell-center, cell volume, a flag iBlank, and interpolation weights. Toaccelerate the search across several intersecting partitions, a pthreads approach is used within each MPIprocess. Thus, instead of using the following approach:

// In each process p

for j : All intersecting partitions not in the grid corresponding to process "p"

associateDonorsAndReceivers()

End

where the search across partitions begins only when the previous search is complete, the following asyn-chronous call is used:



pthread_create(<associateDonorsAndReceivers()>)

EndFor

// Synchronize at the end



pthread_join()

End

The function associateDonorsAndReceivers() does the job of pairing possible donors with receivers. As-suming that the current process ID or partition ID is i, and that receivers are being associated in partition j,fig. 5 explains how this is achieved. A primary check is evaluated to identify whether the receiver cell centerxc is inside the OBB of the donor partition(current partition). If it is not inside, then there is no donor for xcon the current partition (identified by the red lines in fig. 5). If xc is inside the donor OBB, a further checkis made to identify the OBB cell index corresponding to xc. This is established using the direct map(DM)explained in Sec.III. At times, it is possible that although xc can lie inside the donor OBB, it does not haveto physically lie inside the donor partition. Thus we use the inverse map of the OBB cell index to establisha physical cell in the current partition in the vicinity of xc. If the IM results in a negative physical cell,there is no donor in the current partition. If the IM results in a realistic cell xd, a local search is initiatedwith this xc as a starting donor cell. See for example fig. 6a. A line segment is then created between xcand xd. Plausible intersections of this line segment with the faces of xd are computed. The adjacent cell

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Figure 5. Donor search process

whose face has a valid intersection becomes the new donor cell. As the iteration proceeds, the donor cellstarts off with the one marked in yellow, followed by green and finally red. It may happen that the searchline crosses a partition boundary as in fig. 6b. In that case, there are no intersections and a further check isperformed by computing the interpolation weights of the donor cell with respect to xc. As the interpolationweights do not satisfy the boundedness criteria, the search is resumed across partition boundary faces. Theface which yields the minimum distance with xc from a local list of partitioned faces will result in finding anappropriate starting donor cell as in fig. 6c. Since there are no intersections with xc and the interpolationweights will be bounded, the cell in purple will be the correct donor cell. The located donor is then acceptedonly if its volume is less that the volume associated with xc. Other possible conditions are reflected in fig.5. e.g. if there are three cells overlapping at the same location and all three have the same volume, noneof them will interpolate, but solve using the governing P.D.E. Mandatory recipients are an exception in thesense that, their cell volumes are locally manipulated to have higher values so that they always interpolatefrom the next finer grid.

IV.E. Cell Classification

The purpose of the classification step is to collect all the intersecting partitions’ data for a given partition anddecide what type of cell it can be classified as. For example, a given cell P may not have found a donor frompartition J , but might have found a donor from partition K and also might lie inside the hole of a physicalboundary of partition L. In this case, the cell classification routine decides on the cell hierarchy based onthe physical situation. For this particular case, P would be classified as a hole cell as it physically lies insidea solid body. In general hole cells are given the first priority, followed by mandatory interpolation cells andfinally other interpolation cells. Cells which have not been tagged in this process automatically become thefield cells with iBlank = 1. A communication array is generated in this step, and it holds information aboutthe receiver cell, the donor cell, its iBlank, and its partition.

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(a) Starting donor cell(yellow) (b) Search line across partition boundary

(c) Locating the correct donor cell (purple)

Figure 6. The donor search process using the line-walk strategy

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V. Overset-OpenFOAM Coupling Verification - Low Reynolds NumberComputation

This problem showcases a very good application of the overset grid method in OpenFOAM. For smallrigid body motions, OpenFOAM’s default dynamic mesh methodology is capable of simulating such flowswith minimal computational effort. However, for large relative displacements of multiple moving bodies,there is no efficient methodology available to solve such problems in OpenFOAM. To demonstrate how theoverset grid methodology stands out, in comparison with the existing dynamic mesh methods, a suitabletest case involving the relative motion of cylinders from Xu and Wang15 is picked up. The setup for thisproblem is given in fig .7. Both cylinders initially oscillate about their starting location in a quiescent fluid

Figure 7. Problem setup for the relative motion of two cylinders

till t = 16 and then they approach each other at constant speeds at Re 40. The motion of the cylinders isthus governed by the following kinematics:

xlower =

4π sin

(πt4

), 0 ≤ t ≤ 16,

t− 16, 16 ≤ t ≤ 32

xupper =

16 − 4π sin

(πt4

), 0 ≤ t ≤ 16,

32 − t, 16 ≤ t ≤ 32

From the above kinematics, we can infer that the cylinders travel a maximum displacement of about 16Dwhere D = 1 is the diameter for both cylinders. Each cylinder has one grid partition and the backgroundgrid as two partitions. Figures 8,9 show the computed pressure and vorticity contours when the cylinders arenearest to each other and farthest from each other in comparison with the reference results.15 A satisfactorycomparison is obtained for these visualization plots. Further, a comparison of the lift and drag coefficientson the upper cylinder are plotted in fig. 10. Excellent comparisons are obtained between the present oversetcomputations of Xu and Wang.15

VI. Tandem Cylinders at Re 1.66× 105

The grid setup used for the single grid and overset grid simulations are shown in fig. 11. In the near-wallregion, boundary layer grids were created to ensure that y+ ≤ 1 and there were at least 20 grid pointsto resolve the boundary layer. The grid is made up of isotropic cells of size 0.02 − 0.03D (D being thediameter of the cylinder) in the wake of both the cylinders, gap between the cylinders and upstream ofthe front cylinder. This was done to ensure that the grid cell sizes did not affect the accuracy of the LESregion, which exists everywhere outside the boundary layers. Outside these isotropic cell regions, the gridwas smoothly stretched in all the directions. The grid generation was accomplished using field functionsin GMSH. The grid was initially generated in two-dimensions for the URANS simulations. The grids wereextruded in the span-wise direction for the hybrid simulations. The extrusion spacing was set to 0.02−0.03D

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(a) Pressure contours - IBM (Reference computation) (b) Pressure contours - Overset

(c) Vorticity contours - IBM (Reference computation) (d) Vorticity contours - Overset

Figure 8. Pressure and vorticity contours when the cylinders are nearest to each other

to ensure that the cells remains truly isotropic. For the overset case, each cylinder has its own grid, and boththe cylinder grids overlap with each other and also with a background Cartesian grid. The two dimensionalsimulations required around 151000 cells while the three dimensional simulations required roughly 12 millioncells. An additional overhead of 20% in computational time is incurred for the overset computations due tothe relatively low convergence rate of the pressure equation and additional communication compared to asingle grid. All the simulations were performed using the open-source CFD code OpenFOAM. The convectionterm in the momentum equation was discretized using a blended second-order central difference scheme witha maximum of 20% upwinding. The convection term in the transport equation for the turbulent variableswere discretized using bounded second-order schemes. The PISO algorithm was used for the pressure-velocitycoupling. The resulting algebraic equation for all the flow variables were solved using the default algebraic

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(a) Pressure contours - IBM (Reference computation) (b) Pressure contours - Overset

(c) Vorticity contours - IBM (Reference computation) (d) Vorticity contours - Overset

Figure 9. Pressure and vorticity contours when the cylinders are farthest from each other

solvers in OpenFOAM. When the scaled residual became less than 10−12, the algebraic equations wereconsidered as converged. Time marching was performed using Crank-Nicholson scheme. The time step wasmodified dynamically to ensure a constant CFL number of 0.5. Inflow-outflow boundary conditions were usedalong the stream-wise direction while the wall-normal direction had slip boundary conditions and periodicboundary conditions were used along the span-wise direction for hybrid simulations. Along the wall normaldirection, a no slip boundary condition was employed for velocity and zero gradient boundary conditionhas been used for the pressure term. Boundary conditions for k, ω and νt were specified similar to anyviscous-layer resolved RANS simulations. All the simulations were started from potential flow simulationsand computed for 100 eddy turn over times tU0/D to avoid any dependence on initial conditions. Thedependence on the final results on the initial conditions have been reported for certain hybrid models16 and

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(a) Lift coefficient

(b) Drag coefficient

Figure 10. Force coefficients for the flow past two oscillating cylinders

we wanted to avoid this situation.

VI.A. Force comparisons

In table 1, the computed forces on the front and the rear cylinder are presented in comparison with exper-imental data.2 For the RANS computations, both the single and overset grid computations perform in asimilar way which is expected . The hybrid computations perform significantly better for the front cylinderthan the rear cylinder. It was also noted in Gopalan10 that other linear RANS models do not perform verywell and exhibit large discrepancies with experimental data with an error as high as 70%.

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(a) Single grid

(b) Overset grids

Figure 11. Grid setup for the tandem cylinder test case.

VI.B. Instantaneous Flow Structures

A comparison of the Z-component vorticity, turbulent kinetic energy, turbulent viscosity, and stream-wisevelocity for the single and overset grid cases using the RANS model is presented in fig. 12. Compared tothe hybrid computations of Gopalan,10 many of the flow structures are not visible, which is expected of theRANS models. At this spacing ratio, the front cylinder exhibits weak vortex shedding and both the cylindersbehave as one bluff body. Although one is not able to accurately predict these using the non-linear URANSmodel, prediction of the forces is better than linear models as noted in the previous section. For the oversetcases, it is clearly seen that the contours are continuous across the overset boundary. This is a desirable

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Model CD CLrms

Front % Difference Back % Difference Front Back

Experiments 0.64 - -0.2 - - -

Single grid NSST 0.61 -4.6 -0.25 -25 0.011 0.0283

Single grid NUSST 0.63 -1.5 -0.23 -15 0.015 0.036

Overset grid NSST 0.61 -4.6 -0.25 -25 0.012 0.0289

Table 1. Comparison of Lift and Drag Coefficients

feature as this establishes that communication across several partitions that lie on different grids have takenplace as expected. As a final comparison, the stream-wise velocity, turbulent kinetic energy and turbulentviscosity along the center-line in the gap of the cylinders is shown in fig. 13. A very good agreement isobtained for all the three quantities.

VII. Conclusion and Future Study

An implementation of the overset grid framework11 for turbulent flow computations in OpenFOAM hasbeen presented. Based on the work of Gopalan,10 a baseline URANS turbulence model was chosen asit exhibited acceptable agreement with experimental data for the flow past tandem cylinders at a range ofspacing ratios. As a first step towards implementing a hybrid model using overset grids, a non-linear URANSmodel was initially implemented and tested successfully. To gauge the accuracy of the overset coupling, forcecoefficients, flow contours and turbulent profiles displayed excellent comparison with the corresponding singlegrid computations. Further work would involve implementing the hybrid model and verifying the same fordifferent spacing ratios. As the overset framework (OPErA)11 already has a six degree of freedom motionsolver, further studies will also involve studying the wake and vortex induced vibrations of tandem cylinders.

Acknowledgments

The single grid computations were performed on the IBM intelligent cluster at Information TechnologyServices, Union College and the overset grid computations were performed at the Institute of High Perfor-mance Computing(IHPC)-A ∗ STAR, Singapore. The third author would like to thank Tom Yanuklis forassistance in setting up the cluster for performing OpenFOAM simulations.

References

1Zdravkovich, M. M., “Flow induced oscillations of two interfering circular cylinders,” Journal of Sound and Vibration,Vol. 101, No. 4, Aug. 1985, pp. 511–521.

2Neuhart, D., Jenkins, L., Choudhari, M., and Khorrami, M., “Measurements of the Flowfield Interaction Between TandemCylinders,” 15th AIAA/CEAS Aeroacoustics Conferences, American Institute of Aeronautics and Astronautics, Miami, Florida,May 2009, pp. 2009–3275.

3Chapman, D. R., “Computational aerodynamics development and outlook,” AIAA Journal , Vol. 17, No. 12, 1979,pp. 1293–1313.

4Gopalan, H., Heinz, S., and Stollinger, M. K., “A unified RANS–LES model: Computational development, accuracy andcost,” Journal of Computational Physics, Vol. 249, 2013, pp. 249–274.

5Spalart, P. R., “Detached-eddy simulation,” Annual Review of Fluid Mechanics, Vol. 41, 2009, pp. 181–202.6Noack, R. W., Boger, D. A., Kunz, R. F., and Carrica, P. M., “Suggar++: An improved general overset grid assembly

capability,” Proceedings of the 47th AIAA Aerospace Science and Exhibit , 2009.7Noack, R. W., “DiRTlib: A library to add an overset capability to your flow solver,” 17th AIAA Computational Fluid

Dynamics Conference, AIAA, Paper No. AIAA, Vol. 5116, 2005.8Noack, D. B. E. P. R., “FoamedOver: A Dynamic Overset Grid Implementation in OpenFOAM,” 10th Overset Grid

Symposium, 2010.9Abe, K., Jang, Y.-J., and Leschziner, M. A., “An investigation of wall-anisotropy expressions and length-scale equations

for non-linear eddy-viscosity models,” International Journal of Heat and Fluid Flow , Vol. 24, No. 2, 2003, pp. 181–198.10Gopalan, H. and Jaiman, R., “Hybrid URANS Simulation of Flow Past Tandem Circular Cylinders,” Journal of Wind

Engineering and Industrial Aerodynamics, Submitted , 2014.11Chandar, D., “A Parallel Dynamic Overset Grid Algorithm and its Application in OpenFOAM,” International Journal

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12Roget, B. and Sitaraman, J., “Robust and Scalable Overset Grid Assembly for Partitioned Unstructured Meshes,” AIAAPaper , Vol. 797, 2013.

13Karypis, G. and Kumar, V., “A fast and high quality multilevel scheme for partitioning irregular graphs,” SIAM Journalon scientific Computing, Vol. 20, No. 1, 1998, pp. 359–392.

14Geuzaine, C. and Remacle, J.-F., “Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processingfacilities,” International Journal for Numerical Methods in Engineering, Vol. 79, No. 11, 2009, pp. 1309–1331.

15Xu, S. and Wang, Z. J., “An immersed interface method for simulating the interaction of a fluid with moving boundaries,”Journal of Computational Physics, Vol. 216, No. 2, 2006, pp. 454–493.

16Frohlich, J. and von Terzi, D., “Hybrid LES/RANS methods for the simulation of turbulent flows,” Progress in AerospaceSciences, Vol. 44, No. 5, 2008, pp. 349–377.

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(a) Z-Vorticity - Single grid (b) Z-Vorticity - Overset grids

(c) Turbulent kinetic energy -Single grid

(d) Turbulent kinetic energy -Overset grids

(e) Turbulent viscosity - Singlegrid

(f) Turbulent viscosity - Oversetgrids

(g) Stream-wise velocity - Singlegrid

(h) Stream-wise velocity - Over-set grids

Figure 12. Comparison of the instantaneous flow features between single and overset grids

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(a) Stream-wise velocity (b) Turbulent kinetic energy

(c) Turbulent viscosity

Figure 13. Profiles of indicated quantities along the center-line in the cylinder gap

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