The main goals are the development, validation, and application of a fractional step solution method of the time-dependent incompressible Navier-Stokes equations in generalized coordinate systems. A solution method that combines a finite volume discretization with a novel choice of the dependent variables and a fractional step splitting to obtain accurate solutions in arbitrary geometries is extended to include more general situations, including cases with moving grids. The numerical techniques are enhanced to gain efficiency and generality

Rosenfeld, Moshe

English

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MCAT Institute Final Report 90-001
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Development of a Time-Dependent
Incompressible Navier-Stokes
Solver Based on a
Fractional-Step Method
Moshe Rosenfeld
(NASA-CR-187323) DEVELOPMENT OF A
TIME-OEOENOENT INUOMPRESSISLE NAVIER-STOKES
SOLVER _ASED ON A FRACTIONAL-STEP METHOD
Final Report, I Oct. 198B - 15 Feb. 1990
(_CAT Inst.) 11 p CSCL 20D G}/14
N91-13039
Uncles
0309190
September 1990
MCAT Institute
San Jose, California
https://ntrs.nasa.gov/search.jsp?R=19910004326 2020-03-19T20:57:08+00:00Z
MCAT Institute Final Report 90-001
Development of a Time-Dependent
Incompressible Navier-Stokes
Solver Based on a
Fractional-Step Method
Moshe Rosenfeld
FINAL REPORT
FOR THE PERIOD OCTOBER 1, 1988 TO FEBRUARY 15, 1990
September 1990
F
m
MCAT Institute
3933 Blue Gum Drive
San Jose, California 95127
DEVELOPMENT OF A TIME-DEPENDENT INCOMPRESSIBLE
NAVIER-STOKES SOLVER BASED ON
A FRACTIONAL-STEP METHOD
Moshe Rosenfeld
INTRODUCTION
This study has been focused on the development, validation and
application of a fractional step solution method of the time-
dependent incompressible Navier-Stokes equations in generalized
coordinate systems. A solution method that combines a finite-
volume discretization with a novel choice of the dependent variables
and a fractional step splitting to obtain accurate solutions in
arbitrary geometries has been previously developed for fixed-grids,
see Ref. (1).
In the present research effort, this solution method is
extended to include more general situations, including cases with
moving grids. The numerical techniques are enhanced to gain
efficiency and generality. This report summarizes briefly the work
performed during the period October 1, 1988 through February 15,
1990. Additional details on the various aspects of the study are
given in Appendix A.
NUMERICAL ENHANCEMENTS
The fixed grid solution method has been extended to general
moving grids. Errors originating from the discrete approximation of
the time-dependent coordinate system are minimized by satisfying
the discrete geometric conservation laws for the time-varying
computational cells. To improve the efficiency of the fixed grid
case, two versions of the solution method have been coded: (1)
fixed-grid method, (2) moving-grid method.
During the present study, several enhancements of the
numerical method have been introduced. A partial list of the
modifications is given below:
(1) Implementation of more general boundary conditions
implicitly. The allowable boundary conditions are:
(a)
(b)
(c)
Periodic conditions,
Symmetric conditions,
Mixed Dirichlet and Neumann type boundary conditions.
(2) The solution method has been extended to geometrically
singular boundaries for the three types of grid topologies (C, O
or H grids).
(3) A multi-grid Poisson solver has been written and partially
debugged.
(4) Extensive efforts have been made to increase the efficiency of
the method by improved vectorization. Presently, the fixed-
grid method runs at 80 MFLOPS on the CRAY YMP (single CPU)
and about 300 - 400 . 10 -6 CPU sec/mesh-point/time-step are
consumed. The moving-grid code is not yet fully vectorized.
VALIDATION OF THE METHOD
Several additional cases have been solved to validate the
method against other numerical and experimental results. In all the
cases tested so far, good agreement is obtained. The validation
cases include:
.i
Fixed-Grid Case:
(1) Flow in a two-dimensional polar cavity.
(2) Flow in a two-dimensional channel with a fixed constriction
and a time variable pressure gradient.
(3) Flow over a two-dimensional elliptic airfoil with a steady and
pulsatile upstream flow and a high laminar Reynolds number
(Re = 14,300).
(4) Three-dimensional flow in curved ducts, both with rectangular
and circular cross-sections.
(5) Flow over a submarine body at an incidence of 0 o and 20 o.
2
Additional details of these validation cases can be found in
Ref. (2). A brief summary of preliminary results for two
validation cases is given below:
Flow Over an Elliptical Airfoil at a High Reynolds Number
The two-dimensional flow over an elliptical airfoil of
thickness ratio 1:2.91 at 14o angle of attack and a Reynolds number
of 14,300 has been solved to compare with the recent experimental
results. Two cases have been considered, (a) steady upstream flow,
(b) pulsatile upstream flow. In the second case, a sinusoidally
pusatile upstream flow with an amplitude of 5% of the steady part,
and a non-dimensional period of T = 6.86 was simulated. A non-
orthogonal 0-type grid of 161 x 141 mesh points in the radial and
circumferential directions, respectively, has been used. Figure 1
gives the time-evolution of the lift and drag coefficients for the
steady and pulsatile upstream flows (it should be noted that in the
pulsating case, the time is normalized by the period time T). The
analysis of the results and the comparisons with the experimental
results will be reported elsewhere.
Flow Over a Submarine Body at Low Reynolds Numbers
The axisymmetric flow over a submarine body has been
computed for a low Reynolds number of Re = 1000 and a zero angle of
attack. Figure 2 compares the pressure coefficient on the body of
the submarine with the computed results of Ryan (Private
Communication), while Fig. 3 shows the effect of the Reynolds
number on the pressure coefficient. Figure 4 shows the distribution
of the pressure coefficient for an incidence of 20 o. Figure 5 plots,
for the same case, the limiting streamlines (viewed from the rear
end of the submarine) and the particle traces (side view).
Moving-Grid Case:
(1)
(2)
(3)
Flow over a circular cylinder with a moving outer boundary.
Flow in a two-dimensional channel with a moving constriction.
Flow in a two-dimensional cavity with a moving piston.
3
Additional details on these validation cases can be found in
Appendix A.
CONCLUDING REMARKS
In the present study, a fractional step solution method of the
time-dependent, viscous and incompressible Navier-Stokes
equations has been extended, enhanced and validated for both fixed
and moving generalized coordinate systems. The method has been
used to simulate time-periodic vortical flow fields. Investigation
of these flows by novel analysis methods that are being developed by
the author, may advance the understanding of the complex vortical
flow phenomena found in pulsating flows.
The study has demonstrated the capabilities of the present
fractional solution method in simulating accurately complicated
incompressible time-dependent viscous flow fields.
4
REFERENCES
1. Rosenfeld, M., Kwak, D., and Vinokur, M., "A Solution Method for
the Unsteady and Incompressible Navier-Stokes Equations in
Generalized Coordinate Systems," AIAA 26th Aerospace Sciences
Meeting, January 1988, Reno, Nevada, AIAA paper 88-0718.
. Rosenfeld, M., Kwak, D., and Vinokur, M., "A Fractional-Step
Solution Method for the Unsteady Incompressible Navier-Stokes
Equations in Generalized Coordinate Systems," J. Comp. Physics,
submitted after revision, 1989.
5
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cylinder at 14o incidence and Re = 14,300.
6
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body at 0o incidence.
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submarine body at OO incidence.
7
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ORiGiNAL PAGE IS
OF POOR Q,UALn"Y
Fig. 5 - Limiting streamlines and particle traces for the submarine
body at 20 o incidence.
9


Development of a time-dependent incompressible Navier-Stokes solver based on a fractional-step method

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