livrable-maillage-nsmb - ovetset grid theory

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    3rd step : Calculation of the interpolations parameters. In this step, the

    donor cells and the associated weights that compose the interpolation areevaluated for each valid overlapped cell.

    4th step : Calculation of the interpolation values.

    grid 3 (flat)

    grid 1 (airfoil)

    grid 2 (flap)

    a) b)

    A B C

    WNS

    n

    WHC

    nWNS

    n

    WHC

    n

    interpolation

    grid 2(flap)

    grid 1(airfoil)

    d)

    D

    grid 2

    (flap)

    grid 1(airfoil)

    overlapped cellsc)

    A B C D E

    e)

    g(a

    Figure 1: Main steps of the chimera method.

    In the present work, a three-dimensional structured multi-block chimeramethod is implemented in order to simulate the free falling sphere falling ina tube (Fig? 2). A basic chimera method is used and the main works focus onthe increasing of the speed process with a study on each step of chimera algo-rithm. This work allowed to find out a new criteria for a quickly and efficientlyhole cutting process. The developped chimera method is fully automatic with

    few, if not none, of user input and pre-processing.

    Y

    cylindri

    spherical grid

    XY-plane

    YZ-plane

    Figure 2: Representation of the cylindrical grid that discretizes the pipe spaceand the spherical grid representing the sphere.

    2

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    The paper is organized as follows: the section 2, the details of each step of

    the implemented chimera method are discussed. In this section is described thestudy of the bucket method for detection of overlapped cells, the definition of anew criteria for hole cutting algorithm and the justification of the interpolationscheme. In Section 3 we report the results of test cases to validate the accuracyand the efficiency. We find here the results of the flow around a sphere, thenthe flow around the sphere close to a plane wall and, to finish, the results of freefalling sphere in a tube.

    2 The flow solver

    We implemented the chimera method in the flow solver called Navier-StokesMulti-Block (NSMB) [15, 16]. The NSMB code solves the compressible Navier-

    Stokes equations using a finite volume formulation on Multi-Block structuredgrids and parallelized using the Message Passing Interface.

    Among the many discretization schemes present in NSMB, in this study weuse a central scheme with 4th order artificial dissipation [17] for spatial discreti-sation and an implicit dual time stepping solved by Lower-Upper SymmetricGauss-Seidel method (LS-SGS) for time integration. The artificial compress-ibility method [18] is employed to fulfil the incompressible flow.

    The governing equations are the preconditioned unsteady Navier-Stokes equa-tion written in conservative form with dual-time stepping. Using the Cartesiancoordinates (x,y,z), these equations can be expressed in a conservative form asfollows :

    P1

    (W) + I t

    (W) + x

    (f fv) + y

    (g gv) + z

    (h hv) =0 (1)

    The state vector Wand the the inviscid fluxes f, g andh are given by :

    W = (p, u, v, w)T

    f=

    u, u2 +p, uv, uwT

    g =

    v,vu,v2 +p, vwT

    h=

    w,wu,wv,w2 +pT

    where is the density, u v and w are the Cartesian components of velocity, pis the pressure, dt is the physical time step and d the pseudo-time step. Theviscous fluxes fv, gv and hv are given by :

    fv = (0, xx, xy , xz)T

    gv = (0, yx, yy , yz )T

    hv = (0, zx, zy , zz )T

    with the shear stress tensor given by

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    xx = 23

    2 ux

    vy

    wz

    xy =yx =

    vx

    + uy

    yy =2

    3

    u

    x+ 2

    v

    y

    w

    z

    xz =zx =

    w

    x +

    u

    z

    zz =2

    3

    u

    x

    v

    y+ 2

    w

    z

    yz =zy =

    v

    z+

    w

    y

    where is the dynamic viscosity (Stokes hypothesis).The modified unit matrix I and the pre-conditioning matrix P for the pseudo-

    time equation system are given by

    I=

    0 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    and P =

    2 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    The coefficientcan be seen as a relaxation parameter for the pseudo-time

    solution and is defined as

    2 =max( 2minU 2

    , CU

    2)

    with min = 5 and C = 0.05.

    3 The chimera method

    Details and improvements to meet our constraints (free fall body in a confinedspace) are described following the same steps of the description (detection ofoverlapped cells, determination of the type of overlapped cells, ...)

    3.1 Detection of overlapped cells

    The search of overlapped cells is based on a test of inclusion on the coordinates.In order to accelerate the search, the mapping of coordinates in a virtual uniformCartesian grid is performed. This method is well known as thebucket method[19, 20, 21]. An inverse mapping algorithm creates an index array that linkvirtual grid to the real coordinates. The search begins on the virtual grid andit continues on the physical associated grid points. The creation of link array istime consuming but the gain in the search time is significant (at least a factor20).

    This algorithm is well known but the number of the virtual cells ( Nv) is aparameter not clearly defined. A simple test is performed in order to determi-nate the optimum value ofNv in function of the number of real coordinates N.We consider two uniform three-dimensional Cartesian grids (Nx =Ny =Nz =N = 128) overlapping entire the cube 1x1x1 domain. Identical results wereobtained on a coarser and a finer grid. The cpu-time is determined for range1 Nv 256. In addition to the compromise between cpu-time decrease forvirtual grid creation and cpu-time increase for searching the overset, the sizerequired by the link array must be taken into account. The results, represented

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    on figure 3, show that the optimum value for the virtual cell number Nv is the

    half of the number of physical coordinates N.The same test is performed for agrid with tangent hyperbolic distribution and, in this case, the optimum valuecorrespond to N.

    N

    cpu-time

    0 50 100 150 200 250

    100

    101

    102

    10

    10

    10cpu-time of creation of link arraycpu-time for searchsum of cpu-time (creation+search)array size of virtual grid link

    v

    64=N/2 128=N

    Figure 3: Representation of the cpu-time used for the indexation of real coor-dinates in virtual grid, cpu-time for searching overset and the size of the indexbetween both grids versus the number of virtual box Nv in one direction.

    Another issue concerns the points that are inside a solid body. In this case,the cells are not detected as overlapped cells. To remove these cells, two meth-ods are implemented. For simple cases (cylinder, sphere) we use an analytical

    function and for general cases. We compute the dot product of the vector fromthe nearest wall cell center to cell center and the vector of the associated wallnormal vector. If the dot product is negative, the cell is in the solid region,otherwise the cell is out of the solid region. This second method is 30 timesmore expensive than the first one, for the configuration of a sphere in a tube,with 6 millions of cells and 65 000 wall cells.

    3.2 Determination of the type of overlapped cells

    The second step of the method is to define the type of the overlapped cells(calculated, interpolated, hole). In this step, among several overlapped cell,the best candidate to predict the flow state (calculated cell) and to give theflow solution to the other cells (interpolated cells) is chosen. The criterionthat justifies the choice is based on the best resolution of the physical problem.We distinguish four techniques based on the criteria associated to the followquantities : user-defined grid ranking, user-defined cell quality, cell size, cellsize in the wall-normal direction.

    For simple overset, the definition on an overlapped hierarchy of each blockis enough for the cell type selection. All the cells of the highest ranking ofthe grid are defined as calculated cells. This solution is simple and fast butnot suitable for complex overlapped layers where the definition of chimera gridranking by the user is then needed. Based on the fact that the cell size isa criterion indicating the best discretization, Siikonen et al. [21] or Liao et al.[22] justify this choice by assigning thecalculated cellstype to the smallest ones.

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    This criterion is locally adapted, automatic and requires no user input. A third

    criterion is added to the previous one and based on a user-defined cell quality.This way the user can force a specific kind of cell (calculated or interpolated) byassociating to this cell a high quality value. This technique can force cell type byprotecting or immunizing cells but the user input is binding and does not followthe grid movement. The cell size in the wall-normal direction was introducedby Landmann [14]. The selection use the intersection segment between normalnearest wall and cell boundary. This technique is fully automatic and it is basedon an important physical quality : the boundary layer. This test requires tosearch for the nearest wall and to calculate the intersection for each cell. Inorder to find a lighter, accurate and automatic criterion,

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