AlAA 90-1494 A Method for Incompressible Flow Calculations with Application to Several 3-D Problems M. Williams Rockwell International Canoga Park, CA

and Lasers Conference L

June 18-20, 1990 / Seattle, WA 2

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AIM-90-1494

A METHOD FOR INCOMPRESSIBLE FLOW CALCULATIONS WITH APPLICATION TO SEVERAL 3-0 PROBLEMS

- c Morgan Williams

Rockwell International, Rocketdyne Division Canoga Park, California -

A primitive variable method for incompressible flow calculations is introduced. The principle of the method consists of introducing an amount of psuedo-compressibility into a projection scheme to obtain a Helmholtz type expression for a pressure variable. A flow solver based on this method is described. The flow solver and method are verified by computing the turbulent flow in a circular duct with an 180 degree bend and comparing the results to experimental data. Several three-dimensional problems related to the flow through rocket engine components are described and the incompressible flow method is applied to these problems.

A difficulty arises in solving the incompressible Navier-Stokes equations because, unlike the compressible flow equations, the continuity equation is not explicitly related to the pressure. A pressure field that allows the momentum and

d continuity equation to be simultaneously satisfied must be calculated. This involves deriving an explicit relation for the pressure. This is the fundamental problem of incompressible flow calculations. Because of the difficulty in determining the pressure field, some vorticity-based methods have been investigated1,2. The main feature of these methods is the elimination of pressure as a variable. Methods which retain the pressure as a variable are commonly referred to as primitive variable schemes. These schemes are readily adaptable for three-dimensional flow problems. There are two basic classes of algorithms

for the solution of the primitive variable Navier-Stokes equations for incompressible flow. The first type of algorithm solves a Poisson type equation for the pressure or a variable related to the pressure. The continuity equation is indirectly satisfied through the solution of a Poisson equation. The Harlow and Welch’ approach and the projection type (velocity decomposition) methods4t5r6 belong to this class. These methods mimic the physical propagation speed (infinite speed of sound) of pressure disturbances for incompressible flow. These methods require an accurate solution

’- of the Poisson pressure equation since any intermediate solution can lead to numerical

Member of Technical Staff Copyright 0 1990 by Morgan Williams Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

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instability and does not ensure that the velocity divergence is negligible. Theneed to accurately solve the Poisson

pressure equation can impact the efficiency of these methods f o r the calculation of steady-state phenomena. Peric’ points out that most of the computational processing time for these algorithms is spent solving the Poisson equation. The solution of a pressure Poisson equation can require a large number of iterations when a three-dimensional flow is being computed. The second class of algorithm Solves a

hyperbolic type equation for the pressure. These methods are commonly referred to as psuedo-compressibility scheme^.^,^ The psuedo-compressibility method is based on acoustic principles and replaces the continuity equation with the relaxation equation,

atp = -b div Q, (1)

where p is the pressure, Q is the vector of velocity components (u,v,wIT, superscript T denotes transpose, b>O is a convergence parameter (commonly called the psuedo-compressibility), and atp denotes the variation of pressure with respect to a fictituous or psuedo-time parameter t. This is a relaxation equation for pressure, and the velocity divergence free condition is achieved only for atp=O. The addition of a time derivative term to the continuity equation transforms the incompressible equations to a hyperbolic type system that can be solved by standard, implicit time-marching methods. The success of this method depends on the

selection of the parameter b. The large values of b required to approximate the physical pressure wave propagation speed can not be attained due to numerical stability and convergence constraints. P r e c o n d i t i o n i n g l O can be used to equilibriate pressure wave speeds but it does not rectify the difference between the physical and psuedo-compressibility pressure propagation speed (=db). Also, the pressure wave magnitude depends on the size of b (=-bAt div Q ) . Both of these general algorithms have

proven themselves quite useful for flow analysis.11*12,13,14 An alternative pressure algorithm is presented and a three-dimensional flow solver based on this algorithm is developed here. The details of the pressure algorithm development are

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discussed in Williams.15e16 The ability of the method and flow solver to predict 3-D flow is illustrated by computing the turbulent flow in a U-bend duct. The flow solver is also applied to several flow problems related to rocket engine components. These calculations Were performed to gain more understanding about the physical processes at work, and to supplement analytical estimates.

Flow SOL vex f o r 3 -B Incxmo re?&UeL?bM

Goyernins E a a L i Q U s The governing equations are the

Reynolds-averaged, Navier-Stokes equations with an eddy viscosity approximation. It is assumed that the flow i s steady and incompressible. The governing equations are given as,

a p a , (E-E,) +ay (F-F.,) ta, (G-G,) tQP = o

div Q = axu+ayv+azw = G , (2)

( 3 ) where, Q= u, v, w IT, E= i u2, uv, uw iT, F= [ vu, vz, vw iT, G= ( wu, wv, w2 IT, Q ~ = i a,P, ayP. a,P )T

E ~ = Re-liza,u. (axv + ayu), (axw +azU) I T .

F,= Re-li ( 3 , ~ + a,,), 2aYv, (3,~ +~I,v) I T ,

G,= Re-'( caxw + a,,), (ayw + a , ~ ) , 23,~ i T ,

and the viscous fluxes are given as,

where p is the pressure, u,v, and w are the Cartesian mean velocity components,and superscript T indicates the transpose. The time term atQ is added to allow for the relaxation to a steady-state solution. The variables have been nondimensionalized by the appropriate quantities. The inverse Reynolds number Re'l is the sum of an inverse Reynolds number based on a reference length, velocity and a kinematic viscosity, and an inverse Reynolds number based on a turbulent eddy viscosity. The eddy viscosity ut is related to the turbulent kinetic energy, k , and dissipation rate E by Ut= Cpk2/E

( 4 ) where CL is a constant, and k and E are

obtained from standard k-E turbulence model equations." These equations are supplemented by the temperature equation for incompressible flow.

An auxiliary potential function@can be defined such that,'

\ i_ 0. Q"+' = Q""/7 - Q

( 5 ) where ~ g = i axax$, @yay@, aQ,$ i T and the a s s ensure dimensional balance; superscript n denotes the iteration level and superscript n+1/2 indicates an intermediate o r predicted value. The potential @ i s calculated by taking the divergence of Eq. ( 5 ) and assuming that divQ*+'=-n$ rather than zero,

f2 $-diva$ = -divQn'1/7. ( 6 )

This algorithm helps to satisfy the continuity equation. The complete derivation and rationale behind this equation are presented in The key feature of this equation is that it is relatively easy to solve for all types of boundary conditions.

The algorithm for solving Eqs. ( 2 ) and (3) can now be formulated a s a predictor-corrector scheme.6 The momentum equations are solved for a predicted velocity field Q"''/*. The form of the corresponding momentum equation is,

(Qn+l/Z-Qn) /At tax(E-Ev)"*'/2ta (F-Fv) n 4 1 / 2 Y Y L. +a, (G-Gv) .+1/2,~~"+1/2 = 0

( 7 ) where, E " + 1 / 2 = ( u " u " + 1 / 2 u"v"+1/2 u"w"',/2 I T , with similar expressions for the other terms. The intermediate velocity field Q n i 1 l 2

carries the exact vorticity but does not necessarily satisfy mass conservation.6 T O iteratively satisfy mass conservation, the Helmholtz pressure equation is solved,

Q @ - div Q$= - div Q"+l/?,

( 8 ) with,

0, Q".' = Q""/Z - Q

( 9 ) and the next iterate for the pressure is,

Qp"" = Q " + ( ax@,ay+,az$ IT.

(10) An amount of under-relaxation can be utilized in Eq. ( 1 0 ) if necessary.

Ex~re5'slons for G w z e d Coordi nates. The governing equations are transformed

from Cartesian space to generalized nonorthogonal curvilinear coordinates C S , r l , S ) . F o r example, Eq. (2) be~omes,~3r~~,le atQsrtiIg (E"-E" v) +aq (F"-F",) tag (G"-G",) tQ"p=G,

(11)

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where, Q"=JQ, E"-E",= J(E,, (E-E,) +Sy(~-~,) +E,, (G-G,) )T,

€"-F",= Jlflx(E-Ev) +fly(F-Fv) +flz(G-Gv) IT, G"-G",= J(<, (E-Ev) +cy (F-Fv) +c, (G-Gv) I T , and is also. written in the appropriate form, and J is related to the inverse of the transformation Jacobian and is given as J=a (x, y, z) / a (&,fl, <) , and ex, fix. . . are the standard transformation metrics. A finite-difference method on a

non-staggered grid is used to discretize the governing equations. The viscous terms are discretized by using the expressions given in the paper by Han.18 All the dependent and independent variables are stored at the same grid location. An explicit fourth-order pressure dissipation is used on the right-hand-side of Eq.(6) to damp numerical pressure instabilities , I 1 An option is provided for deferred central differencing or first-order upwind differencing of the momentum convection terms. Artificial dissipation is used to suppress the tendency for odd and even point decoupling. The resulting numerical equations are solved in an iterative fashion using a stongly implicit relaxation, lower-upper (LU) type equation solver .19,20 The resulting computer software, written by the author, incorporating this algorithm for three-dimensional incompressible flow is named Eniqma/3di. The computations described in subsequent sections were

- ,

1 calculated on a SUN 4 computer workstation.

A turbulent duct flow is calculated to examine the predictive ablility of the algorithm and Enigma/3di computer code.

The developing turbulent flow in a U-bend of circular cross-section is calculated. The corresponding experiment is described in Azzola et al.*l Figure 1 shows a schematic of the geometry. The flow Reynolds number based on duct diameter is 57,400. The ratio of the inner radius of Curvature to the duct radius is 2.375. Computations were made over the semi-circular cross-section bounded by the plane of symmetry passing through the center of the pipe and bend center. A 59x14~11 nonuniform grid was used to represent the U-bend. An 0-grid was used for the duct cross-section. The duct length upstream and downstream of the bend is 2 and 7.5 diameters, respectively. Starting inlet profiles were obtained by computing the turbulent flow in a pipe over

a distance of 25 diameters. (The friction factor,based on the static pressure drop, for the pipe was calculated to be 0,0212 and compares well with the theoretical result of 0,0211. This indicates that the turbulence model implementation is correct) . The downstream velocity profle was then used as the inlet profile for the computation of the U-bend flow. Deferred central differencing of the

momentum convection terms was used for this calculation. The convergence history is shown in Fig.2. The computed longitudinal velocity profiles at the inlet, at 90 degrees into the U-bend, and at a distance of 2 diameters downstream of the bend are compared to experimental dataz1 in Fig.3. The agreement is good. The calculated secondary flow at 90 degrees into the U-bend is compared to experiment in Fig. 4 . The calculation overpredicts the secondary flow strength away from the duct wall and underpredicts the secondary flow near the wall.

to RockeL3naine C m € l o w

In this section, the flow solver is applied to two different three-dimensional flow problems related to rocket engine components. The analyses were performed to supplement analytical calculations, and to qualitatively detail various flow mechanisms at work. First-order upwind differencing of the momentum convection terms was used. A more detailed analysis of these flows would require refined grids.

Duct Fuel The fuel flowmeter is used in the space

shuttle main engine (SSME) to monitor the fuel mass flow rate into the engine. The mass flow rate is related to the flowmeter turbine rotational frequency by a calibration constant. The calibration constant is sensitive to the incoming velocity profile. The purpose of the computational analysis is to determine the shape of the longitudinal velocity profiles and if the profiles change with engine power level. The computed velocity profiles can be used to analytically determine the various flowmeter torques and calibration constants. The fuel flowmeter is located in the duct

joining the low pressure fuel pump (LPFP) and the high pressure fuel pump (HPFP). The Gomputational model consists of a semi-circular duct with a 90 degree bend and a turbine shaft downstream of the bend, and a symmetry plane passing through the duct and bend center. An outline of the flowmeter computational model is shown in Fig.5. A 67x17~11 nonuniform grid was used to model the geometry. An 0-grid was used for the duct cross-section. Computations were made for three

different engine power levels: minimum ( M P L ) , rated ( R P L ) , and full (FPL). The corresponding Reynolds numbers based on the duct radius are 1.68x107, 2.54x107, and 2.76x107, respectively.

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The calculated pressure coefficient on the outer and inner walls is shown in Fig.6. The pressure coefficient distribution is nearly the Same for all the power levels. The RPL and FPL cases show a slightly higher pressure coefficient in the downstream tangent than the MPL case. Figure 7 shows the nondimensional

longitudinal velocity distribution at the symmetry plane. The velocity profile at the turbine shaft region is made of two distinct regions: a low velocity zone near the meter inner wall, and a high velocity zone near the outer wall. The profiles are nearly identical for all

the computed cases. The velocity profiles and secondary flow strength depend on the radius of curvature of the duct bend; large scale velocity changes should not be evident since the geometry remains the Same and the Reynolds number remains in the turbulent range for each of the computed cases. This result is consistent with the finding by Azzola et al.21 of profile similarity at different high Reynolds numbers for a U-bend duct. The secondary flow on a plane

perpendicular to the symmetry plane is shown in Fig.8. Profiles upstream and near the turbine shaft region are shown in this figure. The distributions ale practically the same for the different power levels. The incoming secondary flow is damped as it flows through the turbine hub region. The secondary flow will affect the angle-of-attack that the meter turbine will see.

3-D Jet Jet IatemCcim Thermal stresses, due to the interaction

of 8 cold and 8 hot jets, occur near the SSME high pressure oxidizer turbopump (HPOTP) inner bellows. The flow geometry is shown in Fig.9. The flow structure is three-dimensional. A Navier-Stokes analysis is required to confirm the results of one-dimensional analytical temperature estimates for thermal stresses on bellows' weld joints. The 3-D jet-jet interaction geometry is

modeled by a 22.5 degree sector. This Sector includes half of a cold jet and half of a hot jet. Use of symmetry/periodic type boundary conditions ensures that the complete 8 cold-8 hot jet configuration is simulated. A 51x17~8 nonuniform mesh is used to discretize the geometry. The flow is turbulent with a Reynolds number of about 1.6~10'. The cold fluid is hydrogen and the hot

fluid is a hydrogen/steam mixture. For simplicity, the computations were carried out for a single component fluid and the flow is modeled as being incompressible. Two different cold-to-hot momentum ratios were used to attempt to account for cold/hot jet density differences. The 3 - D jet-jet flow was calculated for

the two cases listed in Table 1. The first case corresponds to a cold-to-hot mass flux of 0.25 and a cold-to-hot inlet momentum ratio of 2.5. The second case corresponds to a mass flux ratio of 0.31 and a momentum ratio of 4.0.

-

Particle traces on the Symmetry plane are indicated in Fig.10. The calculated hot jet deflection from the vertical is about 18 degrees for case 1 and 29 degrees for case 2. Empirical correlations,Z1 based on the ratio of jet momentum flux, predict 24 degrees and 31 degrees, respectively. The correlations do not account for jet enclosure effects. A pressure gradient due to flow at the exit also contributes to the bending of the hot jet from the vertical. A analytical one-dimensional energy

balance predicts an exit temperature of about 1300 degrees Rankine for case 1 and 1245 deg R for case 2. The computational results are in agreement with this (see Fig.11). The near bellows temperature is about 700 deg R for both cases. Turbulent diffusion of temperature and secondary flow mixing warms the fluid near the bellows.

\

s!JmmuY

A numerical method based on a Helmholtz pressure equation has been successfully developed f o r the computation of three-dimensional viscous incompressible flows. A flow solver based on finite-difference discretization and generalized curvilinear coordinates has been developed and applied to 3 - D flow problems related to rocket engine components. More work can be done to improve the solver. v

Acknowledcreme ntz L

A. Smith and 8. Principie provided flow and geometry information, and analysis for the rocket engine computations. - 1.C.E. Pearson, A Computer Method for Viscous Flow Problems, J. Fluid Mech., vol. 21, pp. 611-622, 1966.

2.M. Hafez, Proceedings of the 9th A I A A CFD Conference, AIAA 89-1966cp, pp. 359-369, 1989.

3.F.H. Harlow and J . E . Welch, Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface, Phys. F l u i d s , vol. 8, pp. 2182-2189, 1965.

4.A.J. Chorin, Numerical Solution of the Navier-Stokes Equations, Math. Comput. vol. 22, pp.745-762, 1968.

5.S.V. Patankar and D.B. Spalding, A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional u Parabolic Flow, Int, J. Heat Mass Transfer, v01. 15 , pp.1787-1806, 1972. v

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6.M. Braza, P. Chassaing, and H . Ha Minh, Numerical Study and Physical Analysis of the Pressure and Velocity Fields in the Near Wake of a Circular Cylinder, J. Fluid

- Mech., vol. 165, pp.79-130, 1986. #. 7.M. Peric, Efficient Semi-implicit Solving Algorithm for Nine-diagonal Coefficient Matrix, Numerical Heat Transfer, vol. 11, pp.251-279, 1987.

8.A.J. Chorin, A Numerical Method for Solving Incompressible Viscous Flow Problems, J. Comp. Physics vol. 2, pp.12-26, 1967.

9.N.N. Yanenko, The Method of Fractional Steps, Springer-Verlag, New York, pp.105-107, 1971.

10.E. Turkel, Preconditioned Methods for Solvina the Incomuressible and Low SDeed Compressible Equations, J. Comp. Physics, vol. 72, pp.277-298, 1987.

ll.C.M. Rhie and W.L. Chow, A Numerical Study of the Turbulent Flow Past an Isolated Airfoil with Trailing Edge Separation, AIAA 82-0998, 1982.

12.G.E. Schneider, Recent Developments in the Discrete Solution of Incompressible Flow Problems, in Advances in Computer Methods for Partial Differential Equations, R. Vichnevetsky and R.S. Steplemov, ieds.), IMACS-1984, pp.165-173, 1984.

13.D. Kwak, J . L . Chang, S.P. Shanks, and S. - Chakravarthy, A Three-Dimensional Incompressible Navier-Stokes Flow Solver using Primitive Variables, AIAA J., vol. 24, pp. 390-396, 1986.

14.5. Yoon, D. Kwak, and L. Chang, LU-SGS Implicit Algorithm for 3-D Incompressible Navier-Stokes Equations with Source Term, AIAA 89-1964-cp, AIAA 9th CFD Conference Proceedings, 1989.

15.M. Williams, A Method for the Calculation of Incompressible Viscous Flow, submitted for publication, 1990.

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16.M. Williams, A Method for the Calculation of Unsteady Incompressible Viscous Flow, submitted for publcation, 1990.

17.5.E. Launder and D.5. Spalding, The Numerical Calculation of Turbulent Flows, Computational Methods in Applied Mech. and Eng., v01. 3, pp.269-289, 1974.

18.T. Han, A Navier-Stokes Analysis of Three-Dimensional Turbulent Flows around a Bluff Body in Ground Proximity, AIAA 88-3766-CP, AI?+A/ASME/SIAM/APS 1st National - Fluid Dynamics Congress Conference Proceedings, 1988.

L

19.H.L. Stone, Iterative Solution of Imnlicit Auvroximation of Multi-dimensional Partial Differential Equations, SIAM J. Numer. Anal., vol. 5, pp. 530-558, 1968.

20.N.L. Sankar, J.B. Malone, and Y. Tassa, A Strongly Implicit Procedure for Steady Three-Dimensional Transonic Potential Flows, ATAd J., v01.20, pp.598-605. 1982.

21.J. Azzola, J.A.C. Humphrey, H. Iacovides, and B.E. Launder, Developing Turbulent Flow in a U-Bend of Circular Cross-Section: Measurement and Computation, ASME J. Fluids Enq., v01. 108, pp.214-221, 1986.

2E.T.Okamoto and K. Enokida. PerDendicular Impingement of Two Turbulent Plane Free Jets, Bull. JSME, Vol. 25, pp.358-359, 1982.

Case

case

D

Fig.1 U-bend duct geometry.

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? - $4 I I I I 0.0 2.0 4.0 6.0 8.0 10.0

iteration *io3 F i q . 2 Convergence h i s t o r y f o r U-bend t u r b u l e n t f l o w c a l c u l a t i o n .

(a ) x/D=-2 0

0.0 0.2 0.4 0.6 0.8 1.0 r

“I (b) 90 deg

9 0

0.0 0.2 0.4 0.6 0.8 1.0 r

W

L

0

W F i g . 3 Measurements21 ( p o i n t s ) and c a l c u l a t i o n ( c o n t i n u o u s l i n e s ) of t h e l o n g i t u d i n a l ( U ) mean v e l o c i t y component a t l o n g i t u d i n a l s t a t i o n s i n a U-bend d u c t - w i t h s t r a i g h t t a n g e n t s ,

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ol 0 - I I I I I 0.0 0.2 0.4 0.6 0.8 1.0

r Fig.4 Measurementsz1 (Fcints) and calculation (continuous lines) of the secondary (v) mean velocity component at 90 degrees into a U-bend duct.

m

./ fuel flowmeter

flowmeter turbine hub

Fi9.5 Schematic of flowmeter hardware.

7

. . 0.0 2

S Fig.6 Flowmeter model pressure distribution on symmetry plane.

I

9 2.5 4.0 5.5

9 2.5 4.0 5.5

X 0 ~

0

I

Ln

2.5 4.0 5.5 X

Fi9.7 Flowmeter model longitudinal velocity distribution on symmetry plane

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momentum ratio=2.5

? 0

upstream of turbine hub

turbine hub region

0.0 0.2 0.4 0.6 0.8 1.0

momentum ratios4.0

hot jet

X Fig.10 Particle traces near symmetry plane Fig.8 Secondary f l o w Strength

285 de? R .'&: ....

cold-to-hot momentum ratio=2.5 ,.- 6,

hot cold-to-hot momentum ratio=4 . O A :a

hot c- gas

cold gas ~.

outline of

3-D computational model

Fig.9 Schematic of HPOTP Inner Bellows.

W

W Fig.11 Perspective view of computed temperature distribution for jet-jet interaction.

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