A MULTI-ZONE HIGH-ORDER FINITE-DIFFERENCE METHOD FOR THE NAVIER-STOKES EQUATIONS

Akil A. Rangwalla* MCAT Inc., Moffett Field, CA

and Man Mohan Rai**

NASA Ames Research Center, Moffett Field, CA

Abstract

A fourth-order accurate finite-difference method for solving the compressible thin-layer Navier-Stokes equations on grids having multiple zones is presented. The convective terms are calculated using fourth-order upwind biased differences, whereas the the viscous terms are evaluated using fourth-order central differ- ences. The scheme is set in an iterative, implicit frame- work. The method preserves its accuracy at the zonal boundaries. Furthermore, a high order interpolation procedure is implemented to transfer information be- tween zones. Results that demonstrate the high-order accuracy of the scheme are included. In particular, a rotor-stator computation for a subsonic axial flow tur- bine is presented and compared with a previous calcu- lation. The high-order calculations clearly capture the flow physics more accurately for a comparable compu- tational cost.

Introduction

High-order-accurate finite-difference and finite- volume methods are finding increasing use in simu- lating complex flow-fields. Methods using high-order upwind-biased differences have been used in the di- rect simulations of turbulent flows and presented in Refs. 1-2. Other approaches in the development of high-accuracy schemes include compact schemes with improved resolution characteristics (Ref. 3), and dis- persion relation approximating schemes (Refs. 4-5). These methods have been proposed for use in compu- tational aeroacoustics and similar methods have been proposed for use in computational electromagnetics (Refs. 6-7). Recently, fourt h-order-accurate finite- difference and finite-volume methods were developed and presented in Ref. 8 for the solution of hyperbolic systems of conservation equations on smooth curvilin- ear grids. The methods were used to solve the Euler equations. Computations of Ringleb-flow and super- sonic flow over a cylinder were included to demonstrate

* Senior Research Scientist, Senior Member AIAA ** Chief, Turbulence Modeling and Physics Branch, Asso-

ciate Fellow AIAA Copyright 01995 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty- free License to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner.

the high-order-accuracy of the methods and the sta- bility of the boundary and near-boundary procedures.

In this paper, an extension of the fourth-order finite-difference method of Ref. 8 for the thin-layer Navier-Stokes equation is presented. The method is also extended for multiple zones along with higher or- der procedures for points near the zonal boundary as well as higher order interpolations to transfer infor- mation from one zone to another. An example cal- culation of the flow through an experimental turbine stage is presented in order to demonstrate the im- proved accuracy of the method. The numerical re- sults obtained from the high-order method are com- pared to those obtained from a standard third-order accurate upwind-biased method (Ref. 9). In Ref. 9, strict third-order accuracy for the inviscid terms was demonstrated only for cartesian grids whereas the vis- cous terms were second-order accurate. The fourth- order method captures the smaller physical scales of the flow. The computational cost per grid point, per iteration of the fourth-order method is comparable to that of the third-order method.

Numerical Method

The numerical method is an iterative, factored implicit algorithm. To describe the scheme, consider the unsteady, Navier-Stokes equations in two dimen- sions

Qt + E z + Fy = R z + Sy (1)

where

and time marching takes the form

1 - P y = UTxy + VTyy + yp- prey

Making the independent variable transformation

r = t

t = a x , Y, t ) (4)

?I = ~ ( 2 , Y, t )

and the thin-layer approximation (Ref. lo), the follow- ing is obtained

where

The vector is given by

Equations (5) and (7) assume that the surface of the body is a t a constant value of q (in incorporating the thin-layer approximation).

The factored, iterative-implicit, spatially fourth- order-accurate algorithm for the left hand side of Eq. 5 is developed in Ref. 8. The method uses a Newton- like iterative technique to solve the nonlinear finite- difference approximations of Eq. 5. An approximate factorization technique (Ref. 11) is used to reduce the bandwidth of the matrix that must be inverted. The iterative implicit technique with third-order-accurate

24Aq 1 (8)

The inviscid numerical fluxes E and F* are given in Ref. 8. The numerical fluxes are obtained using Osher upwinding, where the left and right states at grid half points are obtained by fifth order upwind biased inter- polations. The operators V, A , and 6 are backward, forward and central differences respectively. The vis- cous term S is given by

where the viscous stress terms at points (i, j + 4) are evaluated using fourth-order central differences. For example, the term ( T , , ) ~ , ~ + ~ is evaluated as

where

(-vi,,+2 + 27vi,j+l - 27vij + vi ,j-1) ( ~ q ) i , ~ + + = 24Aq

(10) and furthermore, it is assumed that ut and vt are negligible in accordance with the thin-layer approxi- mation. To obtain high-order accuracy on curvilinear grids, the metric terms in Eq. 10 should be computed using high-order differences. For example, the term

(%)i,j+ +, which is given by

has to be evaluated to fourth-order accuracy. The dif- ferent terms in Eq. l l a are computed using the follow- ing.

j+t = ( - (')i+),j+$ + 27(*)i++,j++

- 27(.)i-+,j+f + (*)i-+,j+ + ) / 2 W

((*)q)ilj++ = ( - (*)i,j+2 + 27(.)i,j+1

Note that the partial derivatives in the < direction re- quires midpoints of grid cells. The midpoints are ob- tained from the original grid by a piecewise bi-cubic interpolation procedure in computational space. Sim- ilar calculations are performed for other metric terms. Additional details are given in Ref. 8.

The Baldwin-Lomax turbulence model is also im- plemented in the code and the final form of the equa- tions are nondimensionalized with respect to a refer- ence pressure and density. Apditiona! details of the derivation of terms such as ~t~ and if^ on the left hand side of Eq. 8 can be found in Ref. 12, whereas the viscous Jacobian M = 2 is obtained for a consis- tent third-order representation.

Boundarv Conditions

The boundary conditions required when using multiple zones can be broadly classified into two types. The first type consists of the zonal conditions which are implemented at the interfaces of the computational meshes. These conditions are used to transfer infor- mation from one zone to another. The second type consists of the natural boundary conditions imposed on the surface and the outer boundaries of the compu- tational domain.

Airfoil Surface Boundary

The boundary conditions imposed on the airfoil surfaces are neslip and adiabatic wall conditions. In addition to the no-slip condition, the derivative of pres- sure in the direction normal to the wall is set to zero. Additional details are given in Ref. 9.

Inlet and Exit Conditions

Two-dimensional nonreflective boundary condi- tions as developed in Ref. 13 were used at the inlet and exit. These boundary conditions assume that the flow near the inlet and exit is linearizable. Further de- tails about the implementation of these conditions for a rotor-stator configurationcan be found in Ref. 14.

Near-Boundary Treatment

A consequence of using a higher order method with the associated larger stencil of grid points is that special procedures are required for points near the boundaries. There are two different requirements for the present method. The first is the evaluation of the left and right states that are needed at the half points to calculate the numerical fluxes. The interpolation procedure has to be modified from the one used in the interior. The other requirement is the evaluation of spatial derivatives.

Fourth-order-accurate interpolations and third- order-accurate finite-difference representations were employed for the inviscid terms (as in Ref. 8). Con- sider the boundary point (1, j). (This point could be either a point on the surface, or a point on the inlet or exit computational boundaries, or on the zonal bound- aries). The inviscid spatial derivative % at the point (2, j) is calculated to third-order, using a

and the left and right states are evaluated using (12)

(QR);,~ = -5Q2,j + 60Q3,j + 90Q4,j - 20Qslj + ~ Q c , ~

128 (13)

Note that the interpolations for (QL) ,j , (QR) a,, and (QL) ;?,j are fourth-order-accurate andare not upwinded. The h iva t ives involving the viscous terms are dropped to second-order a t the surface and zonal boundary points (i, I), and are given by

and so on.

Zonal Interpolations

The present method has been developed with a view to being applied to cases using multiple zones. Information is transferred between overlapping zones at every subiteration, so as to solve for all zones as

a fully coupled system. This is achieved by interpo- lating the flow variables from the interior of one zone

' to the boundary of an overlapping zone. The inter- polations used in past calculations (Ref. 9), have been linear. However, in order to maintain overall accuracy, a higher order interpolation procedure was developed and implemented. Figure 1 is a schematic of a grid and a point Ic', at which some function has to be in- terpolated.

Fig. 1. Schematic of grid with a pivot point

The computational co-ordinates of the grid are (( , 3 ) and their values are given by the indices of the grid points as shown in the figure. In order to ob- tain a high-order interpolation, the computational co- ordinates (tp, qp) of the point Zp have to be evaluated from the underlying grid to high-order accuracy. This is done by using a high-order interpolation given by

(15) In Eq. 15, the computational ceordinates (tp, l )p) are given implicitly and are obtained using a Newton iter- ative technique. The order of accuracy of the method can be different in the two directions. In the present method, iorder ,jorder = 4 in order to get a bi-cubic interpolation. After obtaining the computational co- ordinates (tp, qp), the flow variables are interpolated using the same interpolation function and is given as

Test cases have indicated that the errors using bi-cubic interpolations can be two orders of magnitude lower than the errors in the linear interpolations.

Geometry

In this section, the geometry and grid for a rotor- stator configuration shown in Fig. 2, are presented.

I OUIEn ZONE

Fig. 2. Rotor-stator configuration

The rotor-stator configuration is from an exper- imental turbine described in Ref. 15. This is a low speed turbine which has 22 stator airfoils and 28 ro- tor airfoils. Extensive numerical results,'for this tur- bine have been presented previously (e.g. Refs. 9, 16 and 17). For this computation the rotor geometry was rescaled to have 22 airfoils. This rescaling allows a relatively inexpensive numerical simulation of a one- rotorlone-stator configuration. This figure also shows the boundaries of the zones that were used to discretize the flow domain. The grids corresponding to the dif- ferent zones are presented next.

Figure 3 shows a coarse rendition of the multi- zone grid that was used to obtain the solution pre- sented in this study. For the sake of clarity, only al- ternate grid lines are plotted.

Fig. 3. Grid used for the calculations

The grids in the inner zones are 0-grids and are generated using an elliptical grid generator. These grids overlap the outer H-grids which are generated algebraically. Furthermore, the grids associated with the rotor airfoils (the aft airfoils) translate along with

the rotor airfoils. The dimensions of all the inner O- grids were (151 x 41) whereas the total dimensions of the outer H-grids were (108 x 71) for the stator airfoil and (1 13 x 71) for the rotor airfoil.

Results

Figures 4-5 show the normalized time-averaged pressure as a function of the axial distance along the stator and rotor airfoils. The normalized time-averaged pressure is defined as

where pa,, is the static pressure averaged over one cycle, (pt)inret is the average total pressure at the inlet, pinlet is the average density at the inlet, and VR is the rotor velocity.

The figures indicate, that for this case, the time- averaged pressures obtained from the present fourth- order method are nearly identical to that obtained from the third-order method. Also, both the methods show generally good agreement with the experimen- tal data. This is expected, because for this particular turbine, the unsteady interactions do not significantly affect the time-averaged pressures on the airfoil sur- faces (a result well documented in previous studies, Refs. 9, 16, & 18)

TIME AVERAGED PRESSURE ON STATOR

AXIAL DISTANCE Fig. 4. Time-averaged pressure distribution on the

stator

TIME AVERAGED PRESSURE ON ROTOR

0 1 2 3 4 5 6 7 8 9

AXIAL DISTANCE

Fig. 5. Time-averaged pressure distribytion on the rotor

The difference between the numerical results and the data on the suction side of the rotor airfoil (4.0 5 z 5 7.0) is attributable to the three-dimensionality of the actual flow. This difference disappears with a three- dimensional calculation, a result that has already been documented in Ref. 18.

Figure 6a shows the instantaneous pressure con- tours obtained using the present fourth-order method, whereas Fig. 6b shows the results obtained using the third-order method. The results clearly show that the high-order method is capable of resolving more flow- field details.

Fig. 6a. Instantaneous pressure contours (4t h order)

The contours upstream and downstream of the rotor- stator pair obtained by the fourth-order method (Fig. 6a) show a greater level of unsteadiness than that obtained by the third-order method (Fig. 6b). The variation of

pressure in the wakes is also captured by the fourth- order method and does indicate the superior accuracy of the method.

Fig. 6b. Instantaneous pressure contours (3rd order)

Figures 7a,b show a close-up view of the stator airfoil. The results obtained using the fourth-order method is shown in Fig. 7a whereas Fig. 7b shows the results obtained using the third-order method. It is ev- ident that the high-order near-boundary procedures at the zonal boundaries coupled with a high-order inter- polation method has significantly reduced errors ass* ciated with the zonal boundaries. In the regions where grids overlap, the mismatch between the contours as- sociated with the different grids is greatly reduced.

Fig. 7b. Close up pressure contours (3rd order)

I *

Figures 8a,b show the instantaneous entropy con- tours. The results indicate that the present method is less diffusive. As a consequence, the vortices shed from the trailing edges remain more compact as they are convected downstream (Fig. 8a). The higher-order method is also able to resolve the shed vortices from the trailing edge of the stator airfoil. The original third-order method is unable to resolve and track these vortices.

Fig. Instantaneous entropy contours (4th order)

Fig. 7a. Close up pressure contours (4th order)

Fig. 8b. Instantaneous entropy contours (3rd order)

The instantaneous pressure and entropy contours indicate that the level of unsteadiness captured by the fourth-order method is higher than the third-order method. A large portion of this unsteadiness is at- tributable to the shed vortices from the trailing edges of the stator and rotor airfoils. The frequencies due to vortex shedding are higher than blade passing fre- quency for this case. The fourth-order method can better support these higher frequencies. Hence it is expected that the pressure signal obtained from the fourth-order method will have a larger amplitude with a higher frequency content than that obtained by the third-order method.

One quantitative measure of the level of unsteadi- ness in the flow (for which experimental data is avail- able) is the pressure amplitude on the surface of the airfoils. The pressure amplitudes were obtained by en- semble averaging the unsteady pressure signal over an integral number of blade passages. Once an ensemble- averaged pressure signal is constructed on the surface of the airfoils, the normalized amplitude is then ob- tained from the maximum and minimum of the ensemble- averaged signal. The normalization is given by

where pm,, and pmin are the maximum and minimum pressures that occur in the ensemble-averaged signal, over a cycle, at a given point. The experimental data was obtained in a similar manner.

Figures 9a-c show the pressure amplitudes on the stator airfoils. Figure 9a shows the pressure ampli- tudes obtained from the present fourth-order method, whereas Fig. 9b shows the amplitudes obtained us- ing the third-order method. The legend in the figures denotes the number of blade passages used to obtain the ensemble average. In these figures, the axial dis- tance corresponds to the relative axial distance from the trailing edge of the stator airfoils. The point x = 0 corresponds to the trailing edge. The suction surface corresponds to z 5 0 whereas the pressure surface cor-

responds to z >_ 0. AMPLITUDE ON STATOR (Fourth-&a Mabod)

3

AXIAL DISTANCX

Fig. 9a. Pressure amplitudes on stator (4th order)

AMPLITUDE ON STATOR (Thkd-crda Method)

AXIAL DISTANCE

Fig. 9b. Pressure amplitudes on stator (3rd order)

The figures clearly bring out the differences be- tween the fourth-order results and the third-order re- sults. The pressure amplitudes obtained using the fourth-order method show a stronger dependence on the number of blade passages used to generate the ensemble-averaged signal. This is due to the presence of the higher frequency content in the pressure due to vortex shedding. At first glance, Figs. 9a,b seem to suggest that the computational cost of the high-order method is far greater than the low-order method, since it requires data to be collected over a greater number of cycles in order to obtain an unchanging ensemble

average. It should be noted however, that both meth- ods required an equal number of blade passages to set up the basic interacting flow field that is no longer influenced by the starting transients. The high-order method captures more of the higher frequencies that are not locked to the blade passing frequency. The experimental data was however obtained by filtering out the contribution of these high frequencies by en- semble averaging the signal. It is but natural that a more accurate numerical method would require a sim- ilar procedure.

AMPLITUDE ON STATOR

Fig. 9c. Comparision of amplitudes on stator

Figure 9c shows a comparision between the pres- sure amplitudes obtained using the two methods with an average over 16 blade passages. Both the methods compare with the data equally well and both show a slightly wider large-amplitude region than that found experimentally. It should be recalled that the current calculation is two-dimensional and uses a rescaled ro- tor airfoil that permits a single-rotor single-stator com- putation. A multi-airfoil calculation changes this for the better as reported in Ref. 16.

Figures 10a-c show similar results on the rotor airfoils. Once again, it is seen that the fourth-order method shows a greater variation with the number of blade passages over which the ensemble-average is gen- erated. In these figures, the axial distance corresponds to the relative axial distance from the leading edge of the rotor airfoils. The point x = 0 corresponds to the leading edge. The suction surface corresponds to z 5 0 whereas the pressure surface corresponds to x 2 0. The agreement between the numerical results and the experimental data is not good on the suction side. The suction side amplitude peak is much higher and shifted to the left, which corresponds to a an actual shift to- wards the trailing edge. This is partially due to the rescaling of the rotor airfoil and and partially due to

the two-dimensional approximation of the calculations (Refs. 16, 18, and 19).

AXIAL DISTANCE

Fig. 10a. Pressure amplitudes on rotor (4th order)

AMPLITUDE ON ROTOR (Third-order Method)

AXIAL DISTANCE

Fig. lob. Pressure amplitudes on rotor (3rd order)

AMPUKIDE ON ROTOR

-10.0 -75 -5.0 -25 0.0 25 5.0 7.5 10.0

AXIAL DISTANCE

Fig. 10c. Comparision of amplitudes on rotor

Summary

An unsteady, high-order-accurate method for solv- ing the Navier-Stokes equations on multi-zone curvilin- ear grids has been developed. The integration method is a spatially fourth-order-accurate and temporally third- order accurate, upwind-biased finite-difference scheme that is set in a factored, iterative-implicit framework. The code has been used to simulate subsonic flow through a turbine stage.

The results obtained using the high-order method and those obtained using a t hird-order met hod are compared. It was found that the high-order method captured more of the unsteady aspects of the flow. This is apparent from the amplitudes of the pressure fluctuations and the instantaneous contour plots of the pressure and entropy. Also, the high-order method is more accurate near the zonal boundaries thus elimi- nating the mismatch between contours in the overlap region. Furthermore, the computational cost of the high-order method is comparable to that of the low- order method.

The results of this paper indicate that high-order methods can be used to capture more detailed physics for comparable computational cost as a corresponding lower order method. This opens up several possibil- ities for the application of high-order methods with corresponding high-order zonal boundary conditions. These methods may conceivably be used with much coarser grids to provide the same degree of accuracy as a third-order method, but at a fraction of the com- putational cost. The extension of these methods to three-dimensions have the potential of making three- dimensional aqcoustic calculations in turbomachines within reach. High-order methods would also be natu-

ral candidates for use in large eddy and direct numer- ical simulations for general geometries.