Applied Computational Fluid Dynamics Techniques - Wiley Episode 2 Part 6 pps

17 OVERLAPPING GRIDSAs seen in previous chapters, the last two decades have witnessed the appearance of a number of numerical techniques to handle problems with complex geometries and/or

performed using a three-stage Runge–Kutta scheme with a Courant number of C = 0.6.

The dispersion calculation was run for 485 s of real time (corresponding to the time of a trainentering, halting, exiting the station and the time for the next train to arrive) on a workstationwith the following characteristics: Dec Alpha chip running at 0.67 GHz, 4 Gbyte of RAM,Linux operating system, Compaq compiler Figures 16.11(b)–(e) show the resulting iso-

surface of concentration level c = 0.0001, as well as the surface velocities for time t = 485 s.

Note the transient nature of the flowfield, which is reflected in the presence of many vortices.Deactivation checks were performed every 5 timesteps The tolerance for deactivation was

set to u= 10−3 The usual run (i.e without deactivation) tookT=5,296 sec, whereas therun with deactivation tookT=526 sec, i.e a saving in excess of 1:10 This large reduction

in computing time was due to two factors: the elements with the most constraining timestepare located at the entry and exit sections, and the concentration cloud only reaches this zonevery late in the run (or not at all); furthermore, as this is an instantaneous release, the region

of elements where concentration is present in meaningful values is always very small ascompared to the overall domain Speedups of this magnitude have enabled the use of 3-D CFDruns to assess the maximum possible damage of contaminant release events (Camelli (2004))and the best possible placement of sensors (Löhner (2005))

17 OVERLAPPING GRIDS

As seen in previous chapters, the last two decades have witnessed the appearance of a number

of numerical techniques to handle problems with complex geometries and/or moving bodies.The main elements of any comprehensive capability to solve this class of problems are:

- automatic grid generation;

- solvers for moving grids/boundaries; and

- treatment of grids surrounding moving bodies

Three leading techniques to resolve problems of this kind are as follows

- Solvers based on unstructured, moving (ALE) grids These guarantee conservation, a

smooth variation of grid size, but incur extra costs due to mesh movement/smoothingtechniques and remeshing, and may have locally reduced accuracy due to distorted

elements; for some examples, see Löhner (1990), Baum et al (1995b), Löhner et al (1999b) and Sharov et al (2000).

- Solvers based on overlapping grids These do not require any remeshing, can use

fast mesh movement techniques, allow for independent component/body gridding, butsuffer from loss of conservation, incur extra costs due to interpolation and may havelocally reduced accuracy due to drastic grid size variation; for some examples, see

Steger et al (1983), Benek et al (1985), Buning et al (1988), Dougherty and Kuan (1989), Meakin and Suhs (1989), Meakin (1993), Nirschl et al (1994), Meakin (1997), Rogers et al (1998) and Regnström et al (2000), Togashi et al (2000), Löhner (2001), Togashi et al (2006a,b).

- Solvers based on adaptive embedded grids These do not require remeshing or mesh

movement, but extra effort is necessary to detect the location of boundaries with respect

to the mesh, to refine/coarsen the mesh as bodies move, and to treat properly thesmall elements that appear when moving boundaries cut through the mesh RANSgridding presents another set of as yet unresolved problems; for some examples, see

Quirk (1994), Karman (1995), Pember et al (1995), Landsberg and Boris (1997), Aftosmis and Melton (1997), Aftosmis et al (2000), LeVeque and Calhoun (2001),

del Pino and Pironneau (2001), Peskin (2002) and Löhner (2004)

A large body of work exists on overlapping grids; in fact, a whole series of conferences

is devoted to the subject (Overset (2002–2006)) In a series of papers, Nakahashi and

co-workers (Nakahashi et al (1999), Togashi et al (2000)) developed and applied overlapping

grids with unstructured grids, showing good performance for the overall scheme A general

Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, Second Edition.

‘distance to wall’ criterion was used to assign which points should be interpolated betweengrids For each grid system, the distance to the body walls was computed Any point from agiven grid falling into the element of another grid was interpolated whenever its distance towall was larger than that of the points of the other grid The interpolation information wasobtained from a nearest-neighbour technique.

In the following, the main elements required for any solver based on overlapping gridswill be discussed: interpolation criteria between overlapping grids, proper values of dominantmesh criteria for background grids (those grids that have no bodies or walls assigned to them)and the external boundaries of embedded grids that require interpolation and lie outside thecomputational domain, interpolation techniques for static and dynamic data, treatment ofgrids that are partially outside the flowfield, and the changes required for flow solvers

17.1 Interpolation criteria

Given a set of overlapping grids, a criterion must be formulated in order to select which pointswill interpolate information from one grid to another in the overlapping regions Criteria thatare used extensively include:

- a fixed number of layers around bodies or inside external domains;

- the distance to the wall δ w, whereby points closer to a body interpolate to those that are

farther away (Nakahashi et al (1999)).

The accuracy of any (convergent) CFD solver will increase as the element size decreases

Therefore, it seems natural to propose as the ‘dominant mesh criterion’ the element size h

itself In this way, the point that is surrounded by the smallest elements is the one on which thesolution is evaluated All that is required is the average element size at points This criterionwill not work for grids with uniform element size For this reason, a better choice for the

‘dominant mesh criterion’ is the product of distance to wall and mesh size δ · h.

In what follows, we will denote by s the generalized ‘distance to wall’ criterion given by

which includes all of the criteria described above:

(a) p = 1, q = 0 for distance to wall;

(b) p = 0, q = 1 for element size;

(c) p = 1, q = 1 for a combination of both criteria.

The generalized ‘distance to wall’ described above requires the determination of the closestdistance to the bodies/walls for each gridpoint A brute force calculation of the shortest

distance from any given point to the wall faces would require O(N p · N b ) , where N pdenotes

the number of points and N bthe number of wall boundary points This is clearly unacceptable

for 3-D problems, where N b ≈ N 2/3

p A near-optimal way to construct the desired distancefunction was described at the end of Chapter 2 (see section 2.7) It uses a combination ofheap lists, the point surrounding point linked listpsup1, psup2, the list of faces on theboundarybface, as well as the faces surrounding points linked listfsup1, fsup2 The

key idea is to move from the surfaces into the domain, obtaining the closest points/faces toany given point from previously computed, near-optimal starting locations Except for points

in the middle of a domain, the algorithm will yield the correct distance to walls with an

algorithmic complexity of O(N p log(N p )), which is clearly much superior to a brute forceapproach

17.2 External boundaries and domains

In many instances, the bodies in the flowfield are surrounded by their own grids, which inturn are embedded into an external ‘background grid’ that defines the farfield flow conditions(see Figure 17.1) In order to define a proper ‘dominant mesh criterion’ for the external grid,

the maximum value of s (see (17.1) above) is computed for the whole domain All points belonging to the external grid are then assigned a value of sext= 2smax In order to force aninterpolation for the points belonging to external boundaries of embedded grids (as well as

the points of two layers adjacent to the external boundaries), the value of s for these points is set to an artificially high value, e.g sout= 10smax

:_0 :_1

:_2

Figure 17.1 External boundaries

17.3 Interpolation: initialization

Any overlapping grid technique requires information transfer between grids, i.e some form

of interpolation This implies that for each point we need to see if there exists an element in adifferent grid from which the flowfield variables should be interpolated A fast, parallel way

of determining the required interpolation information is obtained by reversing the question,i.e by determining which points of a different grid fall into any given element A near-optimalway to construct the desired interpolation information requires a combination of:

- a marker for the domain number of each point; and

- an octree of all gridpoints

The algorithm consists of two parts, which may be summarized as follows (see ure 17.2)

Fig-Part 1: Initialization

- Mark the domain number of each gridpoint;

- Construct an octree for all gridpoints;

(a) (b) (c)

Figure 17.2 Interpolation: (a) bounding box; (b) points (octree); (c) retain points – different domain

and inside element

Part 2: Interpolation

- do: loop over the elements

- Obtain the bounding box for the element;

- From the octree, obtain the points in the bounding box;

- Retain only the points that:

Correspond to other domains/grids;

Are located inside the element;

Have a value of s larger than the one in the element;

of more than one mesh/zone This very effective filter may be summarized as follows

Part 1: Initialization

- Determine the outer bounding box of all grids;

- Determine the number of subdivisions (bins) in x, y, z;

- Set lbins(1:nbins)=0

- Set lelem(1:nelem)=0

Part 2: First pass over the elements

- Obtain the material/mesh/zone nr of the element: iemat;

- Obtain the bounding box for the element;

- Obtain the bins covered by the bounding box;

- do: Loop over the bins covered:

lbins(ibin)=-1;

Part 3: Second pass over the elements

- Obtain the bounding box for the element;

- Obtain the bins covered by the bounding box;

- do: Loop over the bins covered:

- enddo

- enddo

At the end of this second pass over the elements, all the elements that cover bins covered

by elements of more than one mesh/zone will be marked aslelem(ielem)=1 Only theseelements are used to obtain the interpolation information between grids

The use of any ‘dominant mesh criterion’ s can produce points that interpolate to and are

interpolated from other grids A schematic of such a situation is shown in Figure 17.3 for a

0is used for interpolating to the outer points

well Although clearly dangerous, it is found that in most cases this will not produce invalidresults The points that are interpolated from other grids and interpolate to other grids in turnare eliminated from the list of interpolating points For more than two grids, care has to betaken in order to eliminate pairwise grid correspondences For example, it could happen that

point i of grid A is interpolated from grid B but interpolates to grid C In such cases, the

points have to be kept in their respective lists

::

0

1

A

Figure 17.3 Interpolating interpolation points

17.4 Treatment of domains that are partially outside

In some instances, the external boundaries of a region associated with a body will lie outsidethe ‘background grid’ A typical case is illustrated in Figure 17.4, where a portion of the

0will be impossible In order to treat cases such

as this, the points that could not find host elements on a different grid are examined further.Any nearest-neighbour of these points that has not found a host is marked as belonging to anexterior boundary, and the distance to body is set to an accordingly high value Thereafter, theinterpolation between grids is attempted once more This procedure, shown schematically inFigure 17.5, is repeated until no further points can be added to the list of exterior points thathave not found a host

17.5 Removal of inactive regions

In most instances, the distance to wall criterion will produce a list of interpolation points that

, an

:_2 :_1

:_3

Figure 17.4 Non-inclusive domains

Ω_1 Ω_1

Ω_1

(a)

(c) (b)

Figure 17.5 Treatment of external domains

internal region associated with a moving body, covers a substantial portion of the ‘background

1 does not need to be updated It is sufficient tointerpolate a number of ‘layers’

Figure 17.6 Removal of inactive inside regions

An algorithm that can remove these inactive interior regions may be summarized asfollows:

- Initialize a point array lpoin(1:npoin)=1;

- For all interpolated points: Set lpoin(i)=0;

- do: Loop over the layers required

- For each point: Obtain the maximum of lpoin and its neighbours;

In this case, the points surrounding currently interpolated points can be marked, and thedistance to wall test to determine intergrid interpolation is only performed for this set ofelements/points Given that the octree for the points would have to be rebuilt every timestepfor moving bodies, it is advisable to use the classic neighbour-to-neighbour element searchshown schematically in Figure 17.7 and discussed in Chapter 13 The marking of points, aswell as the interpolation, can again be parallelized without difficulties on shared-memorymachines

17.7 Changes to the flow solver

The changes required for any flow solver based on explicit timestepping or iterative implicitschemes are surprisingly modest All that is required is a masking array over the points Forthose points marked as inactive or as being interpolated, the increments in the solution aresimply set to zero The solution is interpolated at the end of each explicit timestep, or within

each iteration for LU-SGS-GMRES (Luo et al (1998)) or PCG (Ramamurti and Löhner

(1996)) solvers

Position of Overlap Point

17.8.1 SPHERE IN CHANNEL (COMPRESSIBLE EULER)

The first case consists of a sphere in a channel The equations solved are the

compress-ible Euler equations The incoming flow is at Ma = 0.67 The CAD data is shown in

Figure 17.8(a), where the boundaries of the different regions are clearly visible The surfacegrids and the solution obtained are shown in Figures 17.8(b) and (c) One can see the smoothtransition of contour lines in the overlap region The solution obtained for a single grid casewith similar mesh size is also plotted, and as one can see, the differences are negligible.The convergence rate for both cases is summarized in Figure 17.8(d) Observe that theconvergence rate is essentially the same for the single grid and overlap grid case

17.8.2 SPHERE IN SHEAR FLOW (INCOMPRESSIBLE NAVIER–STOKES)

The second case consists of a sphere in shear flow The equations solved are the ible Navier–Stokes equations The Reynolds number based on the sphere diameter and the

incompress-incoming flow is Re= 10 The CAD data is shown in Figure 17.9(a), where the boundaries

of the different regions are clearly visible The grid in a cut plane, as well as the pressureand absolute values of the velocity for the overlapping and single grid case are given inFigures 17.9(b)–(d) The overlapping and single grids had 327 961 and 272 434 elements,respectively The convergence rates for both cases are summarized in Figure 17.9(e) As in theprevious case, the differences between the overlapping and single grid cases are negligible

The body forces for the single and overlapping grid case were f x = 2.2283, f y = 0.3176 and

f x = 2.2115, f y = 0.3289, respectively, i.e 0.7% in x and 3.4% in y.

(a) (b)

1e-05 1e-04 0.001 0.01 0.1 1

Timestep

Overlap Body Fitted

Figure 17.8 Sphere in channel: (a) CAD data; (b) surface grids; (c) Mach-number; (d) convergence

rates

17.8.3 SPINNING MISSILE

This is a typical case where overlapping grids can be used advantageously as compared to the

traditional moving/deforming mesh with regridding approach A missile flying at Ma = 2.5 at

an angle of attack of α= 5◦spins around its axis The surface definition of the computationaldomains is shown in Figure 17.10(a) The inner grid is associated with the missile and rotatesrigidly with the prescribed spin rate The outer grid is fixed and serves as background grid

to impose the farfield boundary conditions Once the initial interpolation points have beendetermined, the interpolation parameters for subsequent interpolations on the moved gridsare obtained using the incremental near-neighbour search technique The compressible Eulerequations are integrated using a Crank–Nicholson timestepping scheme At each timestep,

the resulting algebraic system is solved using the LU-SGS–GMRES technique (Luo et al (1998, 1999), Sharov et al (2000b)) Figures 17.10(b) and (c) show the grid and pressures in the plane z= 0 for a particular time The overlap regions are clearly visible Note that onecan barely see any discrepancy in the contour lines across the overlap region; neither are thereany spurious reflections or anomalies in the shocks across the overlap region

plane; (d) Abs(vel) in cut plane; (e) convergence rates