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Numerical Analysis of Three-Dimensional Viscous Internal Flows by rma97348

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									.-   4
 NASA Technical Memorandum 100878

 Numerical Analysis of Three-Dimensional
 Viscous Internal Flows
 :NASA-TM-130878)         N U B E E I C A L ANALYSIS i ~ 2                N88-211 16
 (NASA) 10 p                                        CSCL U l b
                                                                 G3/C 1   C 1363 23

 RodrickV. Chima
 NASA Lewis Research Center
 Cleveland, Ohio


 Jeffrey W. Yokota
 Sverdrup Technology, Inc.
 (Lewis Research Center Group)
 NASA Lewis Research Center
 Cleveland, Ohio

 Prepared for the
 First National Fluid Dynamics Congress
 cosponsored by the AIAA, ASME, ASCE, SIAM, and APS
 Cincinnati, Ohio, July 24-28, 1988

                                                            Rodrick V. Chima
                                                      NASA Lewis Research Center
                                                        Cleveland, OH 44135
                                                           Jeffrey W. Yokota
                                                       Sverdrup Technology, Inc.
                                                     (Lewis Research Center Group)
                                                      NASA Lewis Research Center
.                         ABSTRACT
                                                        Cleveland, OH 44135

          A three-dimensional Navier-Stokes code has been de-         centrifugal impeller. Each of these two-dimensional vis-
    veloped for analysis of turbomachinery blade rows and             cous effects can be expected to lead t o secondary flows in
    other internal flows. The Navier-Stokes equations are             three dimensions. Complex geometries, endwall bound-
    written in a Cartesian coordinate system rotating about           ary layers, tip clearance effects, etc. also lead to three-
    the z-axis, and then mapped t o a general body-fitted             dimensional flows in turbomachinery. It is the goal of
    coordinate system. Streamwise viscous terms are ne-               the present work t o begin to predict some of these three-
    glected using the thin-layer assumption, and turbulence           dimensional viscous effects.
    effects are modelled using the Baldwin-Lomax turbulence                 Several steady three-dimensional analyses for tur-
    model. The equations are discretized using finite differ-         bomachinery have been published lately. Among them
    ences on stacked C-type grids and are solved using a mul-         are the work of Dawes3 and Subramanian and Bozzola4.
    tistage Runge-Kutta algorithm with a spatially-varying            Both used Runge-Kutta schemes implemented on sheared
    time step and implicit residual smoothing.                        H-type grids. H-grids are particularly easy t o generate
         Calculations have been made of a horseshoe vortex            and implement for turbomachinery calculations, but suf-
    formed in front of a flat plate with a round leading edge         fer from poor leading-edge resolution. Rai5 has published
    standing in a turbulent endwall boundary layer. Compar-          a notable analysis of unsteady threedimensional rotor-
    isons are made with experimental data taken by Eckerle           stator interaction in an axial turbine. He used a third-
    and Langston for a circular cylinder under similar con-          order accurate upwind implicit scheme and a system of
    ditions. Computed and measured results are compared              patched and overlaid 0- and 11-type grids for good reso-
    in terms of endwall flow visualization pictures and total        lution of viscous phenomena. His analysis has not been
    pressure loss contours and vector plots on the symme-            used for steady flows, however.
    try plane. Calculated details of the primary vortex show                In this paper we describe a numerical method for
    excellent agreement with the experimental data. The cal-         analyzing three-dimensional viscous flows in isolated tur-
    culations also show a small secondary vortex that was not        bomachinery blade passages. The underlying Cartesian
    seen experimentally. The calculations required about 1.6         formulation allows the method t o be applied easily to
    million words of storage and 1.3 hours of CPU time on a          both Cartesian and cylindrical geometries. Stacked C-
    Cray X-MP computer.                                              type grids give good resolution of critical leading-edge
         Calculations have also been made of an annular tur-         regions.
    bine stator that has been tested experimentally a t NASA                A multistage Runge-Kutta scheme i used t o solve
    Lewis. The Mach number ranged from from about 0.21 a t           the finite-difference form of the thin-layer Navier-Stokes
    the inlet to 0.67 a t the exit, and the axial chord Reynolds     equations with a Baldwin-Lomax turbulence model. A
    number was about 1.7 x lo5. Computed surface pressure            spatially-varying time step and implicit residual smooth-
    distributions compare well with measured values at three         ing are used to accelerate convergence of the scheme to a
    span-wise locations. The endwall boundary layers pro-            steady state. Two calculations are presented to validate
    duce horseshoe vortices a t the leading edge of the blade.       the analysis. The first calculation shows the formation
    Computed wake profiles resemble the measured profiles,           of a horseshoe vortex at the leading edge of a flat plate
    but computed efficiencies are lower than measured values         with a round leading edge that stands in a turbulent end-
    by 3 factor of two.                                              wall boundary layer. Comparisons made between calcu-
                                                                     lated results and experimental data for a circular cylin-
                                                                     der under similar conditions show excellent agreement
                      INTRODUCTION                                   between static pressure distributions and flow visualiea-
                                                                     tion pictures on the endwall, and between static and total
         Much of our work in the past few years has in-              pressure contours and velocity vector plots made on the
    volved the analysis of two-dimensional or quasi-three-           symmetry plane. The second calculation is of the flow
    dimensional blade-to-blade flows in turbomachinery'*2.           through an annular turbine stator. Comparisons made
    In these two references, both Euler and Navier-Stokes re-        between calculated and measured static pressure distri-
    sults were presented for each blade row considered, and in       butions compare well at three span-wise locations.
    each case a significant viscous effect was observed. These
    effects included such things as a pressure-surface or 'cove'
    separation on an axial turbine blade, a reduction in the                      GOVERNING EQUATIONS
    choking mass flow and a change in shock location for an
    axial compressor blade, and a reduction in the peak Mach                The Navier-Stokes equations are written in a Carte-
    number and shock strength near the leading edge of a             sian   ( 2 ,y, z ) coordinate system rotating with angular
                                                                              ORIGINAL PAGE IS
                                                                              OF POOR QUALITY.

velocity R about the axis. Th rotatic introduces                    and
source termn in the y- and z- momentum equations. The
Cartesian equations are mapped to a general body-fitted
                                                                                   Cl = q::    + rl,' + Or2
(E, q ,<) coordinate system using standard techniques.                                    q
                                                                                   c,= 3 rlrafJu+ rluaov + Vratlw)
The (-coordinate direction is assumed t o follow the flow                          c = rlrfz + rlvs, +
                                                                                    3                         VSSS
direction and the thin-layer approximation is used to
drop all viscous derivatives in this direction. All v s i-                         c = $ ( < z a r U + < y a r V + <za,w)
cous t e r n in the cross-channel (q,s) plane are retained,                        (35 = tlraru + qvarv + qrarw
with an option in the code to delete all crossderivatives                                                                                (74
if desired. The resulting equations are as follows.                                       1
                                                                    Terms multiplied by C and C2 lead to non-mixed second
                                                                    derivative viscous terms like u,, while terms multiplied
                                                                    by C 4 5 lead to mixed-derivative terms like uVr. The
                                                                    viscous flux vector 8 can be written similarly, with di-
where:                                                              rections 1 and < everywhere interchanged.
                                                                         Metric terms are defined wing the following rela-

                                                                               [:; ;; :I

                                                                                 L      rlr    sr

                                                                               1 Yo% - Yr%

                                                                                                 Yr-? - Y V r
                                                                                 z ~ z , , z,,z~ ~ € 2- Z ~ Z ( Z,,.ZE
                                                                                        - ZrYo

                                                                                                    zrY( - ZQYr

                                                                                                                               - YvzC
                                                                                                                               - ZCZ,,
                                                                                                                               - ZvYt    1   (8)

The velocities in 4 are absolute with respect to the co-
ordinate system fixed to the blade. Relative velocities            (Z(YrjZ,   + ZrYtz, + ZoYrzC - ZtYrz,             - ZnYtZr - Z < Y , ~ C ) - ~
(denoted by a prime] are given by:                                                                                                       (9)
                                                                         The equations are nondimensionahed by arbitrary
                                                                    reference quantities (here the inlet total density pore/
                        u' = u                                      and the total sonic velocity core! were used,) and the
                        v'=v-rlnZ                                   Reynolds number Re and Prandtl number P r are defined
                                                                    in terms of these quantities. The equations assume that
                        w' = w + Ry                                 the specific heats C, and C, and Prandtl number are
                                                    (3)             constant, that Stoke's hypothesis is valid, and that the
and the relative contravariant velocitiy components are             effective viscosity for turbulent flows may be written as
given by:
                                                                                              Pe/l = Plam     + Pturb                    ( 10)
                 U' = Ezu + EvY' +        EZW'
                                                                    where the laminar viscosity is calculated using a power
                 V' = q,u + qyvf + q,w'                             law function of temperature:
                 W' = <=u + <,v' +        {,W'

Note that although u' = u, U' # U.
    The energy and static pressure are given by:
                                                                    with n =    3 for air.
            e=p    [CJ + (u2 + Y2        + w')/2]        (5)

          p = (7- 1) [e    - p(u'     + v z + w2)/2]     (6)                         TURBULENCE MODEL
    Using Stoke's hypothesis, A = - $pl the viscous flux                 The Baldwin-Lomax algebraic two-layer eddy viscos-
Pv can be written as follows:                                       ity model6 is applied on cross-channel (11, <) planes. Two                      I
                                                                    modifications to the standard model are made to account
           pv   = J - ' P [ O , F2,   41F4,      F5IT   (7 4        for the endwall boundary layer and the blade boundary
                                                                    layer and wake, and their interaction in corners.
                                                                         First, the distance from the wall is calculated using
                                                                    the Buleev' length scale d:

                                                                    where s , and sr are normal distances from the walls in
                                                                    the q- and <- directions respectively. This length scale

has the desirable property that d approaches the normal                  For subsonic outflow the exit static pressure is spec-
distance from one wall a t large distances from the other           ified and ( p , pu, pu, p w ) are extrapolated. For Cartesian
wall.                                                               geometries the exit pressure is constant. For annular
     Secondly the turbulent viscosities are calculated              geometries the hub pressure is specified and the radial
across each boundary layer or wake separately, then the             pressure distribution is found by integrating the axisym-
total turbulent viscosity is taken as the vector sum of the         metric radial momentum equation:
components, Le.,
                                                                                     dP - PVe2 - -P v z -
                                                                                                   (           wy)2
                                                                                     dr    r     13

                                                                         Sidewalls and the trailing-edge cut are treated as
This assumption has the desirable properties that outside           periodic boundaries.
of one viscous layer p t u r b takes on values calculated for            On the blade surface V' = 0, and for viscous flows
the other layer, that it goes to sero in the core flow, and         U' = W' = 0. Blade surface pressures are found from the
that near corners it accounts for both walls.                       normal momentum equation. On the hub (e = 1) and tip
                                                                    ( 5 = 5rnaz):

             COMPUTATIONAL GRID                                        W   z   +5 v L +c * W € P +   (fzrlz   + evvv + Szrl*)a,P
     Two-dimensional body-fitted grids for this work
were generated using the GRAPE code developed by
Sorenson '. Threedimensional grids were formed by
stacking the 2-D grids. Figure 1 shows a 3-D grid around                                                                           . ,
a plate with a round leading edge. For annular geome-               On the blades (r) = 1) the normal momentum equation
tries the 2-D grids were stacked along a radial stacking            can be found from (18) by replacing everywhere by r)
line and stretched in the r)-direction so that the blade            and V' by W'.
shape remained constant and the angular pitch of the
outer (periodic) boundary remained constant.
                                                                    MULTISTAGE RUNGE-KUTTA ALGORITHM

            BOUNDARY CONDITIONS                                          The governing equations are discretined using a
                                                                    node-centered finite difference scheme. Second order cen-
     A t the inlet, total temperature T re/ is specified as
                                        o                           tral differences are used throughout.
a constant. A cdistribution of total pressure (PolPo ref)                The multistage Runge-Kutta scheme developed by
is specified, c/2/2- as a constant or as appropriate for an         Jameson, Schmidt, and Turkel' is used to advance the
inlet boundary layer with given thickness and a power-              flow equations in time from an initial guess to a steady
law velocity profile. For Cartesian geometries the (2,    y)        state. If we rewrite (I) as
and (2, ) flow angles are specified. For cylindrical ge-
ometries the (2, y) flow angle is replaced by the inlet whirl
     For supersonic inlet flows, all flow variables are spec-       where Rr is the inviscid residual including the source
ified a t the inlet. For subsonic flows the inlet condi-            term, RV is the viscous residual, an D is an artificial
tions are updated each iteration by extrapolating the               dissipation term described in the next section, then the
upstream-running Riemann invariant R - based on the                 multistage Runge-Kutta algorithm can be written as fol-
absolute total velocity Q = du2+ v 2 w 2 to the inlet.              lows:
                                                                             90 = Qn
                                                                             91 = Qo - aiJAt[Rr Qo - (Rv D ) QO]  +

The total velocity is then found from the    TO     using:

                                                                    For efficiency both the physicial and artificial dissipation
                                                                    terms are calculated only a t the first stage, then are held
                                                        (15)        constant for subsequent stages.
Velocity components are found from Q and the specified
angles or whirl. Within the endwall boundary layer, that
is, where Po/Porcf < 1, the v and w velocity components
are found by extrapolation from upstream. The density                           ARTIFICIAL DISSIPATION
is found using:
                                                                         The dissipative term D in ( 2 0 ) is a nonconservative
                                                                    vereion of that used by Jameson et al.' It is given by:

                                                                                     DQ = (4 o
                                                                                           +D           + D,)Q                (224

where the (direction operator is given by                             Jameson". The technique involves replacing the residual
                                                                      calculated in (20) with a value that has been smoothed
                Dcq = c (v29cc - VrPcctc)                 (22b)       by an implicit filter, Le.,

                            c=-                           (224
                                 JAt                                  where 6,c, 6,,,, and 6,, are standard second difference
in a coefficient that cancella similar terma in (21). To              operators and E € , e,, and E , are smoothing parameters.
minimize the artificial dissipation in viscous regions we                  Linear stability analysis has shown that the Runge-
reduce C linearly acroas several grid points t o cero at the          Kutta scheme with implicit residual smoothing may be
walls.                                                                made unconditionally stable if the E smoothing parame-
                                                                      ters are made sufficiently large. In one dimension
      The terms V and V. are given by:
                                                                                        € 1f [ ( $ ) 2 - l ]               (25)

                                                                      gives unconditional stability if A' is the Courant limit
where                                                                 of the unsmoothed scheme, and X is a larger operating
                                                                      Courant number. In three dimensions different E'S may
                                                                      be used in each direction, and their magnitudes may be
and                                                                   often be reduced below the value given by Eq. (25.)
                           P2 = O(1)
                           P = O(&)
                           .                                                                 RESULTS
In smooth regions of the flow the dissipative terms are of                 Two sets of computed results are presented for pre-
third order and do not detract from the formal second-                liminary validation of the code described above. The first
order accuracy of the scheme. Near shocks vi,j is large               set of results shows the structure of a horseshoe vortex
and the dissipative terms become locally of first order.              formed a t the base of a cylinder standing in a turbulent
                                                                      boundary layer. The second set of results is for turbu-
                                                                      lent flow through an annular turbine cascade. Computed
 THREE-DIMENSIONAL STABILITY LIMIT                                    results are compared to experimental data in each case.
                                                                           When a boundary layer approaches a local obstruc-
     Applying a linear stability analysis to the inviscid             tion such as the leading edge of a turbine blade, the low
form of (20-21) gives the following expression for the time           momentum fluid in the boundary layer often cannot over-
step.                                                                 come the local pressure gradient and the flow separates
                                                                      from the wall. In front of the obstacle the separation
                                 CFL                                  creates a vortex which convects around the sides of the
   At 5
          I,lul+ l u p l   + l E p l+ c d g q T i j T =               obstacle and forms a characteristic horseshoe-shaped flow
                                                          (234             Eckerle and Langston" have made detailed mea-
where                                                                 surements of the horseshoe vortex in front of and around
                                                                      a cylinder of diameter D centered between the sidewalls
                   L = I L I + lllzl+ lfIl                            of a wind tunnel. Test conditions included an inlet Mach
                   1, = I&lI+ l l l Y l + Ifvl                        number of 0.084, R C D= 5.5 x lo5,and an upstream tur-
                   1, = ILI + lllrl+ If*(                             bulent boundary layer thickness 6 = 0.lD. Detailed sur-
                                                     (23b)            face flow visualization and static pressure measurements
The Courant limit for a particular multistage scheme de-              were made, and static and total pressure measurements
pends on the number of stages and the choice of coeffi-               were taken using a five-hole probe.
cients ui. See Ref. 9 for several examples.                                Figure 1 shows the grid used to compute Eckerle
     To accelerate convergence to a steady state we use               and Langston's flow. The grid shown has been coars-
the maximum permissible time step a t each grid point so              ened for clarity. The actual grid had 65 x 49 x 25 points
that the Courant number is constant everywhere. The                   with an initial spacing a t the walls Asi = 0.0010. To
time step is calculated once based on the initial condi-              avoid questions of trailing-edge vortex shedding, a tail     4
tions. It is stored and is not updated during the calcula-            board waa added from the back half of the cylinder to
tions.                                                                the exit boundary. The base grid is approximately 6D
                                                                      square and .5D high, t o match the dimensions of Eck-
                                                                      erle and Langston's wind tunnel test section. We used a
      IMPLICIT RESIDUAL SMOOTHING                                     symmetry condition at mid span but computed the full
                                                                      symmetric flow side-to-side.
     Residual smoothing was introduced by Lerat (see                       The experimental inlet Mach number of 0.084 is too
for example Ref. 10) for use with the Lax-Wendroff                    low for the compressible algorithm used here, s the cal-
scheme and was later applied to Runge-Kutta schemes by                                            i
                                                                      culations were run with M , = 0.2. The peak Mach


                                                                                       DR,IGHNAI; PAGE IS
                                                                                       OF POOR QUALITY
    number was 0.34 at the cylinder-plate junction, so the              junction. There is excellent agreement between the com-
    flow was essentially incompressible. The Reynolds num-               puted and experimental data.
    ber and inlet boundary layer thickness were matched to                    The velocity vectors show how the flow rolls up to
    the experimental data.                                               form a horseshoe ahead of the cylinder. Experimental
         Even with Min = 0.2 the four-stage Runge-Kutta                  velocity vectors are missing in areas where the flow an-
    scheme seemed t o converge poorly, so we eventually ran              gle exceeded the calibrated range of the five-hole probe.
    the calculations with a two-stage scheme with ai =                   From this data Eckerle and Langston12 concluded that
    (1.2, l.),CFL = 4., and implicit residual smoothing at               “The reverse flow did not roll up t o form a vortex,
    each stage with e t = 2., e,, = cr = 4. The initial conver-         however. The vectors clearly show that a closed vor-
    gence rate was fast, as shown in Fig. 2 by the histories            tex was not present in the plane of symmetry, though
    of the maximum and r m s residuals, but after 1000 iter-            positive pitch angles in a portion of the reverse flow at
    ations the solution showed only a tiny horseshoe vortex             R I D = 0.72 and Y/D= 0.02 may indicate the start of
    that did not match the data. Over the next 3000 iter-               vortex formation. Rather than rolling up, flow passed
    ations the residuals changed little, but the vortex grew            out of the plane and proceeded tangentially around the
    and moved upstream until it stabilized a t the position             cylinder.”
    shown later. The total solution took 1.3 hours on a Cray                  The computed vectors clearly show a vortex in the
    X-MP computer.                                                      symmetry plane. A small counterrotating secondary vor-
         Figures 3 and 4 compare the calculated and experi-             tex is also shown a t the cylinder-endwall junction. The
    mental data on the endwall. Figure 3 shows contours of              dimension of the secondary vortex is about 1.5 times the
    constant static pressure coefficient,defined by:                    diameter of the five-hole probe, and would have been
                                                                        nearly impossible to detect experimentally.
                                                                              The second set of results is for an annular cascade
                                                                        of constant profile turbine stator vanes developed and
                                                                        tested a t NASA                   The annular ring has 36
    Computed contours are on the bottom and measured                    vanes with a hub-tip radius ratio of 0.85 and a tip di-
    contours are on the top. The calculations show excel-               ameter of 508 mm. The vanes themselves are 38.10 mm
    lent agreement with the data from the symmetry plane                high and have an axial chord of 38.23 mm. Design flow
     to about 45 degrees around the cylinder, where the in-             conditions are for a fully axial inflow with a hub-static to
    fluence of the tailboard becomes apparent. The lower                inlet-total pressure ratio of 0.6705. These conditions cor-
     part of Figure 4 shows a flow visualization picture made           respond to average inlet and exit Mach numbers of 0.211
     with ink dots on the endwall. It clearly shows the sepa-           and 0.665 respectively. The Reynolds number based on
     ration line and reverse flow region ahead of the cylinder.         axial chord is 1.73 x lo6.
    The computed vector plot at the top of the figure shows
                                                                              A grid consisting of 97 x 31 x 33 points with an initial
    close agreement with the measured separation line loca-
                                                                        spacing a t the wall of 0.0002 of a blade chord was used
    tion and flow directions. The vectors are one point o f   f
                                                                        for the flow calculations and is shown in Figure 7.
    the endwall, and are all drawn to the same length, so
    they show direction only.                                                The calculation was run with a four-stage scheme
          Figure 5 compares the static pressure coefficient dis-       with ai = (1/4, 1/3, 1/2, 1) and C F L = 5.5, using
    tributions on the endwall along the symmetry line ahead            implicit residual smoothing after each stage with e t =
    of the cylinder. The experimental data (circles) show              e,, = et = 0.75. Convergence histories for the annular
    a general pressure rise upstream due to the cylinder               cascade calculation are shown in Fig. 8 where the log of
    blockage, but also a large dip in the pressure inside the          the maximum and rms-averaged residuals have dropped
    separated region. Two vertical bars indicate substan-              approximately 3.5 orders of magnitude in 1500 iterations.
    tial unsteadiness in the experimental data. Eckerle and            The total CPU time was approximately 54 minutes for
    Langston included a 2-D potential solution (solid line)            this calculation.
    for comparison. The computed solution (line with trian-                  Mach number contours a t mid span are shown in Fig.
    gles) shows good agreement with the data ahead of the              9 to illustrate the blade boundary layer and wake thick-
    saddle point, but falls short of predicting the magnitude          nesses. There are approximately 12 grid points across
    of the pressure dip. The discrepancy may be due to the             the pressure surface boundary layer.
    unsteadiness in the real flow or to lack of resolution in                The inlet boundary later thicknesses were specified
    the computed solution.                                             as 1.9 percent span on the hub and 7.1 percent span on
          Figure 6 compares computed (bottom) and experi-              the tip, corresponding to the measured data in Ref. 13.
    mental (top) velocity vectors and total pressure loss co-          In Fig. 10, velocity vectors with superimposed contours
    efficient contours on the symmetry plane upstream of the           of total pressure show how these boundary layers roll up
    cylinder. The total pressure loss coefficient is defined by:       into horseshoe vortices at the leading edge of the blade.
                                                                       The primary vortices are considerably smaller than the
                                                                       inlet boundary layers, and each primary vortex has an
                                                           (27)        even smaller counterrotating secondary vortex associated
                                                                       with it.
    These contours show nearly horizontal boundary layer-                    Figure 11 shows a comparison between the calcu-
    like flow upstream that rolls up into a vortex with a high         lated surface static pressure distribution and data ob-
    loss core. The low loss fluid above the boundary layer             tained by Goldman and S e a ~ h o l t s ’ ~ locations of 13.3,
    curves down the face of the cylinder and carries high              50.0, and 86.7 percent span. The calculated results agree
    momentum fluid to the region near the cylinder-endwall             very well with the experimental data a t all spanwise lo-

cations. Although the blade section is constant from hub                  It is felt that with additional work and experience
to tip, the increased pitch at the tip increases the loading         this analysis will prove to be a useful tool for investigat-
of the tip section considerably.                                     ing three-dimensional viscous flow phenomena in turbo-
     Figure 12 compares calculated (top) and measured                machinery.
(bottom, Ref. 14) efficiency contours on a cross-channel
surface located 1/3 axial chord downstream of the trail-
ing edge. The kinetic energy efficiency is theoretically                                 REFERENCES
independent of the axial position and is defined by the
following:                                                            I. Chima, R. V., “Inviscid and Viscous Flows in Cas-
                    v =QLal/QLeal                                        cades with an Explicit Multiple-Grid Algorithm,”
                                                                         AIAA Journal, Vol. 23, No. 10, Oct. 1985, pp.
where                                                                    1556-1563.
                                                                      2. Chima, R. V., “Explicit Multigrid Algorithm for
              Q%,, = 2CpTo 1 - -
                                (     T ;a)
                                       a: ‘                                 Quasi-Three-Dimensional Viscous Flows in Turbo-
                                                                            machinery,” J. Propulsion and Power, Vol. 3, No.
and   TOis taken as constant.                                               5, S e p t . 4 c t . 1987, pp. 3 9 7 4 0 5 .
      The efficiency contours clearly delineate the endwall           3.    Dawes, W. N., “A Numerical Analysis of the Three-
boundary layers and wake. The computed wake is thin-                        Dimensional Viscous Flow in a Transonic Com-
ner than the measured wake and has higher losses (lower                     pressor Rotor and Comparison With Experiment,”
efficiency) a t the center. Integrating the efficiencies over               J. Turbomachinery, VoL 109, Jan. 1987, pp. 83-90.
the entire area gives a total efficiency of 0.960 for the real        4.    Subramanian, S. V., and Boccola, R., “Numeri-
machine and 0.923 for the calculations. The discrepancy                     cal Simulation of Three-Dimensional Flow Fields
could be due to inadequate resolution of the thick, round                   in Turbomachinery Blade Rows Using the Com-
trailing edge, an inadequate turbulence model, or possi-                    pressible NavierStokes Equations,” AIAA Paper 87-
bly to unsteady vortex shedding in the real flow. The                       1314, June, 1987.
high calculated losses seem not to be due to numerical                5.    Rail M. M.,“Unsteady Three-Dimensional Navier-
dissipation. Indeed, the computed wake appears to be                        Stokes Simulations of Turbine RotorStator Interac-
less dissipative than the measured wake, and the com-                       tions,” AIAA Paper 87-2058, June, 1987.
puted loss is remarkably insensitive to the artificial vis-           6.    Baldwin, B. S., and Lomax, H., “Thin-Layer Ap-
cosity coefficient. Considerable work is clearly needed in                  proximation and Algebraic Model for Separated Tur-
the area of modelling the flows leaving the blunt trailing                  bulent Flows,” AIAA Paper 78-257, Jan. 1978.
edges commonly found on turbomachinery blades.                        7.   Gessner, F. B., Po, J. K., ‘A Reynolds Stress Model
                                                                           for Turbulent Corner Flows - Part I1 Comparisons
                                                                           Between Theory and Experiment,” J. Fluids Enur.,
            C 0NCLUDING REMARKS                                            June, 1976, pp. 269-277.
                                                                      8.   Sorenson, R. L., “A Computer Program to Generate
      A numerical analysis has been developed for three-                   Two-Dimensional Grids About Airfoils and Other
dimensional viscous internal flows. The analysis solves                    Shapes by the Use of Poisson’s Equation,” NASA
the 3-D NavierStokes equations written in a general                        TM-81198, 1980.
body-fitted coordinate system, including rotation about               9.   Jameson, A., Schmidt, W., and Turkel, E., ”Numer-
the z-axis. The thin-layer approximation is made in the                    ical Solutions of the Euler Equations by Finite Vol-
streamwise direction but all viscous terms are included                    ume Methods Using Runge-Kutta Time-Stepping
in the cross-planes. The Baldwin-Lomax eddy-viscosity                      Schemes,” AIAA Paper 81-1259, June 1981.
model is used for turbulent flows.                                   10.   Hollanders, H., Lerat, A., and Peyret, R., “Three-
     An explicit multistage Runge-Kutta scheme is used                     Dimensional Calculation of Transonic Viscous Flows
to solve the finitedifference form of the flow equations.                  by an Implicit Method,” AIAA Journal, Vol. 23, No.
A variable time step and implicit residual smoothing are                   11, Nov. 1985, pp. 1670-1678.
used to accelerate the convergence of the scheme. Con-               11.   Jameson, A., and Baker, T. J., “Solution of the Eu-
vergence rates are slow a t low Mach numbers but reason-                   ler Equations for Complex Configurations,” AIAA
able for typical turbomachinery applications. We hope                      Paper 83-1929, July 1983.
to improve convergence rates by adding multigrid to the              12.   Eckerle, W. A., and Langston, L. S., “Measurements
code.                                                                      of a Turbulent Horseshoe Vortex Formed Around
     Results showing the development of a horseshoe vor-                   a Cylinder,’ NASA Contractor Report 3986, June,
tex in a turbulent boundary layer ahead of a cylinder are                  1986.
presented to validate the analysis. Excellent agreement              13.   Goldman, L. J., and Seasholtc, R. G., ‘Laser
is found between the computed results and experimental                     Anemometer Measurements in an Annular Cascade
data for this case.                                                        of Core Turbine Vanes and Comparison With The-
     Results for an annular turbine cascade also show                      ory,” NASA Technical Paper 2018, June, 1982.
horseshoe vortex development. Very good agreement was                14.   Goldman, L. J., and McLallin, K. L., “Cold-Air
found between measured and computed surface pressure                       Annular-Cascade Investigation of Aerodynamic Per-
distributions. Computed losses were high and show the                      formance of Core-Engine-Cooled Turbine Vanes. I:
need for improved modelling of flows around blunt trail-                   Solid-Vane Performance and Facility Description,”
ing edges commonly found in turbomachinery-.                               NASA T M X-3224, 1975.

  MACH   0.196             RE    538257.   ALPHA        0.00           IIEP    WOO

 'igure 1. Computational grid for the horseshoe vorte,

                                                                                                  rcxrar     I LAYCSIOYS no+?sfsmr         voerrx E x p r e i m r N I                  I- 2
  NlCH     0.196          PC         538261.   1iPHO      0.00         IICP    to00               I4Cn      0.196     RE    538257.           ALPXA        0.00             Iff#        WOO
  CPS C D U I D U P Z     "I*        -2.000    rrx        .0
                                                         I00           IUC    0.IOO

Figure 3. Comparison of measured (top) and computed                                            Figure 4. Comparison of endwall flow visualization (bot-
(bottom) static pressure contours on the endwall.                                              tom) and computed flow direction vectors (top) ahead of
                                                                                               the cylinder.
                                                                                                        I     ,     1     1     1      I        I      1       1   -    1          I      1     I

         05         06          01        08      09         10         11     12     I:


Figure 5. Comparison of measured and computed static
pressure distributions on the endwall symmetry line.                                             rcxrsir     I LANGSIONS n o n s f w o r   vosfrx    rXPrPiwY7                         I = 33
                                                                                                 NACH       0.196     er      538257.        ~ P N A    0.00                /ire        woo
                                                                                                 c p r coxrooas         "iu    o.000         "AX       0.600                INC        0.100

                                                                                               Figure 6. Comparison of measured (top) and computed
                                                                                               (bottom) velocity vectors and total pressure loss contours
                                                                                               on the symmetry plane.





  G O L O l l N S A U Y U L l P CASCAOEr 3-0

                                                                                            Figure 8. Convergence history for the annular turbine
cascade.                                                                                    cascade problem.


                                                                                                                 - _

   GOLONAYS I N Y U L A P   c*scmr. 3-0                                      I - 18            SOLOCANS ANNULAR CASCAOC. 3 - 0
   x ~ c n 0.21~
                                                                                                                P I   173000.
                                                                                                                                      1LPnA    0.00
                                                                                                                                                             o.aa2   1
Figure 9. Mach number contours a t mid-span.

            0 . h
       L?:       " Y Y U L I P C*SC*OE.

               0.212         PC
                                      l.11000.   hLPnA    0.00   IIrP
                                                                       .llP6 X .
                                                                                   It           GOIOM1YS IYUULAP C4SCAOI, 3-0
                                                                                                XlCl   0.212

Figure 11. Comparison of measured and computed pres-                                          pigure 12. Comparison of measured (bottom) and com-
sure distributions a t three span-wise locations.                                            puted (top) efficiency contours 1/3 axial chord down-
                                                                                             stream of the trailing edge.

                                                                                        a                        ..               j.2 2--.q--g3
                                                                                                                        -   1
                                                                                                                                  . :?-u,uT-y
    I         Nallonal Aearonaulcr and
              Space M m i n i m r a l m
                                                                  Report Documentation Page
         1.    Report No.                                                2.   Government Accession No.                             3. Recipient's Catalog No.
                NASA TM-I 00878
         4. Title and Subtitle                                                                                                     5.   Report Date

                N u m e r i c a l A n a l y s i s o f Three-Dimensional V i s c o u s
                I n t e r n a l Flows                                                                                              6.   Performing Organization Code

         7.    Author@)
                R o d r i c k V. Chima and J e f f r e y W . Yokota
                                                                                                                                   8. Performing Organization Report No.
                                                                                                                                         E-41 12
                                                                                                                                  10. Work   Unit No.
         9. Performing Organization Name and Address
                                                                                                                                  11.   Contract or Grant No.
               N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n
               Lewis Research C e n t e r
    1          C l e v e l a n d , O h i o 44135-3191                                                                             13. Type of Report and Period Covered

112.           Sponsoring Agency Name and Address                                                                             I          T e c h n i c a l Memorandum
               N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n                                        14.   Sponsoring Agency Code
               Washington, D.C.                  20546-0001

        15.    Supplementary Notes
               P r e p a r e d f o r t h e F i r s t N a t i o n a l F l u i d Dynamics Congress cosponsored b y t h e A I A A ,
               ASME, ASCE, S I A M , and APS, C i n c i n n a t i , Ohio, J u l y 24-28, 1988. R o d r i c k V. Chima,
               NASA Lewis Research C e n t e r ; J e f f r e y W . Yokota, S v e r d r u p Technology, I n c . , ( L e w i s
               Research C e n t e r Group), NASA Lewis Research C e n t e r , C l e v e l a n d , O h i o 44135.

        16.   Abstract                                                                                                                                                   -
               A three-dimensional                    Navier-Stokes code has been developed f o r a n a l y s i s o f t u r b o m a c h i n e r y b l a d e rows
                and o t h e r i n t e r n a l f l o w s . The Navier-Stokes equati,ons a r e w r i t t e n i n a C a r t e s i a n c o o r d i n a t e system
                 r o t a t i n g a b o u t t h e x - a x i s , and then mapped t o a g e n e r a l b o d y - f i t t e d c o o r d i n a t e system. Streamwise
                v i s c o u s terms a r e n e g l e c t e d u s i n g t h e t h i n - l a y e r assumption, and t u r b u l e n c e e f f e c t s a r e m o d e l l e d
                u s i n g t h e Baldwwin-Lomax t u r b u l e n c e model. The e q u a t i o n s a r e d i s c r e t i z e d u s i n g f i n i t e d i f f e r e n c e s on
               ' s t a c k e d C-type g r i d s and a r e s o l v e d u s i n g a m u l t i s t a g e Runge-Kutta a l g o r i t h m w i t h a s p a t i a l l y - v a r y i n g
                 t i m e s t e p and i m p l i c i t r e s i d u a l smoothing. C a l c u l a t i o n s have been made o f a horseshoe v o r t e x formed
                 i n f r o n t o f a f l a t p l a t e w i t h a round l e a d i n g edge s t a n d i n g i n a t u r b u l e n t e n d w a l l boundary l a y e r .
                Comparisons a r e made w i t h e x p e r i m e n t a l d a t a t a k e n by E c k e r l e and Langston f o r a c i r c u l a r c y l i n d e r
                under s i m i l a r c o n d i t i o n s . Computed and measured r e s u l t s a r e compared i n terms o f e n d w a l l f l o w
                v i s u a l i z a t i o n p i c t u r e s and t o t a l p r e s s u r e l o s s c o n t o u r s and v e c t o r p l o t s on t h e symmetry pJane.
                C a l c u l a t e d d e t a i l s o f t h e p r i m a r y v o r t e x show e x c e l l e n t agreement w i t h t h e e x p e r i m e n t a l d a t a . The
                c a l c u l a t i o n s a l s o show a s m a l l secondary v o r t e x t h a t was n o t seen e x p e r i m e n t a l l y .           The c a l c u l a t i o n s
                r e q u i r e d a b o u t 1 . 6 m i l l i o n words o f s t o r a g e and 1.3 h o u r s o f CPU t i m e on a Cray X-MP computer.
1               C a l c u l a t i o n s have a l s o been made o f an a n n u l a r t u r b i n e s t a t o r t h a t has been t e s t e d e x p e r i m e n t a l l y a t
                NASA Lewis Research C e n t e r .                 The Mach number ranged rom a b o u t 0.21 a t t h e i n l e t t o 0.67 a t t h e e , i i t ,
                and t h e a x i a l c h o r d Reynolds number was a b o u t 1 . 7 ~ 1 0 . Computed s u r f a c e p r e s s u r e d i s t r i b u t i o n s
                compare w e l l w i t h measured v a l u e s a t t h r e e span-wise l o c a t i o n s . The e n d w a l l boundary l a y e r s produce
                horseshoe v o r t i c e s a t t h e l e a d i n g edge o f t h e b l a d e . Computed wake p r o f i l e s resemble t h e measured
                p r o f i l e s , b u t computed e f f i c i e n c i e s a r e l o w e r t h a n measured v a l u e s by a f a c t o r o f two.

    17. Key Words (Suggested by Author@))                                                          18.    Distribution Statement
               Navier-Stokes equations                                                                    Unclassified - Unlimited
               T h r e e - d i m e n s i o n a l flow                                                     S u b j e c t C a t e g o r y 01

    19. Security Classif. (of this report)                       20.   Security Classif. (of this page)

                            Unclassified                                              Unclassified                                               10                    A02

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