AI AA-87-0543 Navier-Stokes Simulations of Blade-Vortex Interaction Using High-Order Accurate Upwind Schemes M. M. Rai, NASA Ames Research Center, Moffett Field, CA

AIM 25th Aerospace Sciences Meeting January 12-15, 1987/Reno, Nevada

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NAVIER-STOKES SIMULATIONS OF BLADE-VORTEX INTERACTION USIX HIGH-ORDER ACCURATE UPWIND SCHEMES

Man Mohan Rai* NASA Ames Research Center, Moffett Field, California v

Abstract

Conventional, spatially second-order- accurate, finite-difference schemes are much too dissipative for calculations involving vortices that travel large distances (relative to some measure of the size of the vortex). This study presents a fifth-order-accurate upwind-biased scheme that preserves vortex structure For much longer times than existing second-order-accurate central and upwind difference schemes. calculations demonstrating this aspect of the fifth-order scheme are also presented. The method is then applied to the blade-vortex interaction problem. the vortex impinges directly on the airfoil o r a Shock associated with the airfoil are presented. None of these calculations required any modeling of the shape, size, and trajectory of the inter- acting vortex.

Vortex

Results for strong interactions wherein

Introduction

Many commonly encountered flow fields in applied aerodynamics involve the interaction between vortices and airfoils. A typical example

-I is the helicopter rotor where the advancing blades encounter the tip vortices of preceding blades. A similar situation exists in turbomachinery where the tip vortex which is generated in the tip clearance region between the rotor and the shroud interacts with the downstream stator airfoils. In addition, unsteady loading of upstream rows of airfoils results in unsteady vortex shedding, and these vortices interact with downstream airfoils. Vortex-blade interactions, in turn, result in unsteady loading of airfoils and noise. fore, a clear understanding of vortex-blade interaction is necessary in order to predict the unsteady loads on airfoils and noise levels. a predictive capability is crucial to improved blade designs and noise minimization.

There-

Such

Previous work in computing vortex-blade interactions has ranged from solving the simple, small-disturbance equation to the highly complex compressible, thin-layer Navier-Stokes equa- tions. Reference l presents results for two- dimensional blade-vortex interaction using the small-disturbance equation. The vortex is modeled by introducing a branch cut in the grid across which a jump in the potential is specified. The motion of the vortex is effected by stepping the

*Research Scientist. Member AIAA.

This paper is declared a work of the U S Government and ./

therefore is in the public domain.

edge of the branch cut through the grid. ence 2 presents unsteady three-dimensional results botained using the full potential equations. presence and motion of the vortex is simulated with a "split-potential" formulation wherein the total potential is assumed to be a sum of a known potential (corresponding to the vortex), and an unknown potential. The known potential gives rise to source terms in tne governing equations which simulate a moving vortex in the flow field. The computed results are compared with the experi- mental results of Ref. 3 and the two are found to be in fairly good agreement.

Refer-

The

Reference 4 presents unsteady, thin-layer Navier-Stakes results for a two-dimensional analog of the real three-dimensional configuration of Ref. 3. An analytically defined Lamb-type vortex is released several chords upstream of the air- foil, and its progression through the grid and interaction with the airfoil flaw field is calcu- lated using the Beam-Warming ~ c h e m e . ~ One of the problems with such calculations, where the vortex is calculated as a part of the solution and not imposed on the solution, is that conventional, second-order-accurate, finite-difference schemes possess enough artificial dissipation to smear and to otherwise distort the vortex very rapidly. Since vortex initialization must necessarily be done Several chords upstream of the airfoil (because of the slow decay of the vortex velocity field with distance), the shape and strength of the vortex are distorted considerably by the time it has traveled close to the airfoil. In general it can be said that the coarser the finite- difference grid that is used to perform the cal- culation, t h e greater the distortion of the vortex. In order to perform an accurate blade- vortex simulation with existing schemes, an extra fine grid would have to extend all the way to the vortex initialization region. Such a calculation would be prohibitively expensive. Reference 4 uses the "perturbation" approach, wherein a priori knowledge of the position, shape, and strength of the vortex can be used to generate source terms in the governing equations that counteract the effect of the artificial dissipation of the numerical scheme. The computed results of Ref. 4 are com- pared with the experimental results of Ref. 3. The agreement between the two sets of results is found to be good.

Reference 6 also presents results obtained with the unsteady Navier-Stokes equations and the Beam-Warming scheme. An alternate approach is used to solve the vortex distortion problem. The numerical dissipation term (a fourth derivative of the dependent variables) is evaluated using the difference between the total solution and the flow

1

field associated with the vortex. This approach removes the dissipation caused by the vortex structure, and preserves the vortex for longer lengths of travel. However, in order to subtract the vortex solution from the total solution, one needs to know the shape, strength, and position of the vortex as in Ref. 4.

Altnough Refs. 4 and 6 do not impose a known vortex flow field on the airfoil calculation a s in Refs. 1 snd 2, they do require some knowledge of the vortex i n order to negate the effects of excessive numerical dissipation. For this reason, the approaches used in Refs. 4 and 6 are restricted to "weak interaction" cases where the vortex is at a moderate distance from the airfoil throughout its travel. In cases where the vortex comes close to the airfoil (say less than 0.25 chord), the trajectory and shape of the vortex ctiange considerably; hence, they cannot be specified a priori. In fact, i n some cases the vortex may split into two o r more vortices. Obvi- ously a more general proceduri than thGSC used in Refs. 4 rind 6 is required to solve the more diffi- cult problems associated with close encounters.

The most straightforward approach to the blade-vortex interaction problem is to use a numerical scheme which is much more accurate than the conventional second-order accurate schemes, and whose leading dissipative term is a couple of orders smaller. in magnitude than those of the second-order schemes. Such a high-order accurate scheme can routinely be used fo r both strong and weak interactions, and w i l l be the most likely candidate algorithm for a general purpose rotor code. Since the perturbation approach of Ref. 4 is independent of the scheme used, it can also be used with the more accurate method to enhance the accuracy of the calculation in cases where suffi- cient information regarding the vortex is available.

The use of higher-order accurate methods alleviates the numerical dissipation problem; however, there will always be a lower limit on the number of grid points required per core radius of the given vortex. Hence, as the core radius of the vortex gets smaller, the number of grid points that are required for the calculation becomes greater. In situations where the trajectory of the vortex is approximately known (to within a chord o r two) special high density grids uith the required nuniber of grid points can be patched into coarser grids. The technology to transfer infor- mation from grid to grid in an accurate manner has been developed in Refs. 7-9.

Whereas both central and upwind difference Schemes can be modified to be more accurate, it may be worthwhile to choose upwind schemes for this purpose. The reasons for this choice are manifold, some of which are given below. First, central difference schemes require the use of arbitrary smooth parameters to stabilize them. The choice of these parameters is an art that has

been developed over many years. A new learning process would be necessary to establish the right smoothing parameters for the new scheme. Upwind schemes on the other hand do not require the spe- cification of such arbitrary parameters. Second, upwind schemes model the physics of the problem m o w accurately, and can be expected to yield norc accurate solutions.

W

In this study ire present a fifth-order 1cc11- rate upwind scheme that is set i n an iterative implicit framework. The scheme is second-order accurate in time and can be made to solve the nonlinear, fully implicit, finite-difference equa- tions corresponding to the Navier-Stokes equations at each time-step. The scheme is dcrived with the Osher flux-differencing approach. but it car1 bc used in conjunction with other types of flux dif- ferencing such as Roe's scheme.ll The new scheme is then tested by calculating the motion of a vortex in a free str'eam and monitoring a measlire of the rate of decay of the vortex. ity of this method as compared to conventional

case. The new method is then applied to the blade-vortex interaction problem. The blade- vortex code is first validated for weak interac- tions using the experimental data of Ref. 3 and the numerical data of Ref. 4 . The code is then used to simulate strong blade-vortex interactions such as "head-on" collisions. The number of grid points required for the calculation is minimized by patching a fine grid into the region of travel of the vortex. Information transfer between L patches is performed using the methods developed in Refs. 7-9.

The supcrior-

second-order methods i S demonstrated in this

The Integration Method

Both upwind and central difference schemes have been and are currently being used extensively to obtain solutions to fluid-flow problems. 'These two approaches have their relative merits and demerits. Central difference schemes are simple to program, require relatively few arithmetic operations at each grid point, and are ineXperlSiVE to use. Unfortunately, they require the specifi- cation of some arbitrary smoothing parameters and they are sensitive to the choice of these parme- ters; thus, they loose some of the robustness that is required of numerical schemes in order to build reliable, general purpose C Q ~ ~ S . On the othe? hand, upwind schemes are more complicated, require more computation per grid point, and are more expensive to Use. However, they model the physics more accurately and do not require the specifica- tion of any arbitrary smoothing parameters. For these reasons, upwind schemes may be the appropri- ate choice in building reliable, general purpose codes.

One criticism of upwind schemes is that

used variety) possess dissipation terms that a r e much larger than those found in conventional,

second-order upwind schemes (the mast commonly I/

second-order, central-difference schemes. In fact, the Beam-Warming central-difference scheme5 does not have any natural dissipation terms; the only dissipation terms are those that are added

'...-. artificially to stabilize the scheme. The larger amounts of dissipation that conventional second- order upwind schemes exhibit may adversely affect Navier-Stokes calculations because the dissipation of the scheme may be of the same magnitude as the natural viscosity of the fluid (in many situations central-difference schemes require large amounts of smoothing to stabilize them, and in no way better resolve the viscously dominated regions of the f low). In order to overcome this difficulty, specially constructed, low-truncation-error, second-order- and third-order-accurate, upwind schemes are proposed in Ref. 12. The relevant truncation error term in these schemes is still a third-order, fourth derivative, but it is multi- plied by constants that are smaller in magnitude; hence, the overall dissipation is about the same 3s that used commonly in second-order, central- difference schemes. Whereas the modifications proposed in Ref. 12 result in upwind schemes that are adequate for viscous calculations, these new schemes are not sufficiently accurate to simulate vortex motion over several core radii.

Typically, in using upwind schemes, most of the computational expense is incurred in calculat- ing the flux differences between grid points. Once these f l u x differences are calculated, they can be used to construct upwind schemes of an arbitrarily high order of accuracy without increasing computational costs significantly. The integration method presented in this study is a fifth-OPder-accurate, upwind-biased scheme which uses the Osher approach in calculating the flux differences between grid points. The leading truncation error term for this scheme is a fifth- order, sixth derivative; hence, this scheme can be expected to yield solutions that are far superior to those that can be obtained with any second- order-accurate, central-difference scheme. The scheme is developed within the framework of an iterative implicit method.13 first-order- and second-opder-iterative implicit methods of Ref. 13 are first described, and then the extensions necessary to increase the order of accuracy to fifth-order are outlined.

The First-Order-Accurate Scheme

--

In what follows the

To describe the first-order-accurate scheme we consider the unsteady Euler equations in two dimensions,

The vectors Q, E, and F are given by

OU

ouv

e + p

where 0 is the density; p i s the pressure; u and v are the velocities in the x - and y-directions, respectively; and e is the total energy per unit volume.

2 e = + 0 (u2 + v ) v - 1 2

Under the independent variable transformation

r = t

5 = <(X,Y,t) ( 2 )

n = ,l(X,Y,t)

E q . (1) transforms into

0 7 + E 5 + F n r O

where

(3)

a = Q/J

E ( Q , < ) = (ItQ + 5,E + 5 F)/J

F(Q.n) = (ntQ + nxE + n F)/J Y

Y J = 5 n - n 5 X Y X Y

The notation t (Q,<) and F(Q,n) is used to show the dependence of these quantities an the metrics of the transformation.

A conservative finite-difference scheme for E q . (3) can be written as

0:: - oyd Ei+,,* "m , - Em 1-1/2d

a r + 65

where the i:+1,2, and i: +,,2 are numerical fluxes Consistent dith the'dransformed fluxes and P, respectively. [ E q . ( 4 ) l is explicit when m = n and is fully implicit when m i n + 1 .

order-accurate Osher scheme 1s b?ven by

The difference scheme

The numerical flux E . for the first- 1+1/2

3

where

where

Details regarding the evaluation of the integral in Eq.-(6) can be found in Ref. 10. The numerical flux Fi,j+1/2 can be obtained using a similar expression.

An evaluation of the numerical flux in Eq. ( 5 ) using the dependent variables at the n t h level results in an explicit scheme and dif- ference equations that are linear. However, an evaluation of these fluxes at the (n+l)th time levei results in an implicit scheme and difference equ'itians that are nonlinear and need to be solved in an iterative mame?. The usual strategy that is employed at this stage is the linearization of the numerical fluxes with respect to the time-like variable, i. The resulting system of linear equa- tions is then solved in order to update the depen- dent variables. The linearization process depends on the scheme used to determine the numerical fluxes. The linearization of the Osher fluxes is discussed in Ref. 13.

Using a Newt n type of linearization on Eq. ( 4 ) with an+ as the independent variable, we obtain the iterative implicit technique of Ref. 1 3 as applied to the first-order-accurate Osher scheme

? -

and d a n d P are forward and backward difference operators, respective y In Eq. ( 7 ) bP is an approximation to 0 , When p = 0, 0' c 0" and when Eq. ( 7 ) is iterated to convergence at a z i v m time-step, ap = 0"". because the left-hand side of this equation can be made equal to zero at each time-step (by iterating to convergence), linearization errors can he driven to zero during the iterative process. For problems where only the asymptotic steady-state is of interest, the iteration process need not he carried to convergence at each time-step. I n fact, when the number of iterations is restricted to one, the scheme reverts to a conventional non- iterative scheme of the type in Ref. 5 (hut unfactored).

" + I '

It Should be noted that,

Unfortunately, Eq. ( 7 ) is extremely time consuming to solve in a direct fashion because of the large bandwidth of the matrix on the left-hand side. At this point two options are available to the user: the first option is to use approximate factorization as in Refs. 5 and 13; the Second option is to use a relaxation strategy as in Refs. 14 and 15. The approximately factomd form of Eq. ( 7 ) is given below.

P [ I + - ( V 5 A+ 1,j + "AI ,J ) ]

= RHS of equation ( 7 ) (8 )

Clearly, approximate factorization reduces the bandwidth (the single large-bandwidth matrix is Converted into two block-tridiagonai matrices). If the iteration process is carried to Conver - gence, the factorization error is driven to zero. However, at very large time-steps, f a c t o r i - zation error may cause the iteration process to diverge or to enter a limit c y c l e .

The Second-Order-Accurate Scheme

The numerical flux Ef+i,*,j for the second- order-accurate Osher Scheme is given by

L

W

- AE+(Q. .,pi+, , j,ti+l12, ? 1.J

where the AE' ape evaluated as before. Lineari- zation of all the terms in Eq. ( 9 ) would result in block-pentadiagonal matrices after approximate factorization. Hence, only the terms correspond- ing to the first-order scheme are linearized. ?he resulting iterative implicit scheme takes the form

+ ?,.j+l/2 - ?9 (lo) A"

where P = yi?. One of the differences between Eq. ( 8 ) and Eq. (10) is that the first-order- accurate, finite-difference used for the time derivative of 6 on the right-hand side of Eq. (8) has been replaced with a second-order- accurate, finite-difference. The Newton lineari- zation of this term a l so reflects the change in accuracy. This modification in addition to the new numerical fluxes [Eq. ( 9 ) 1 results in a scheme that is second-order accurate in both space and time. As before, in order to achieve time- accuracy, Eq. (10) needs to be converged at each time-step.

Fully Upwind Schemes Versus Upwind-Biased Schemes

Before extending the Second-order-accurate scheme to a fifth-order-accurate scheme, it is instructive to consider one particular aspect of upwind schemes, that is, the size of the stencil

-

or the number of consecutive grid points required to achieve a certain order of accuracy. In order to do this, we consider the simple wave equation in one dimension

Ut + u 2 0 (1 1 )

The first-order-accurate, iterative, implicit scheme given by Eq. (81, as applied to Eq. ( l l ? , reduces to

n n r l n + l 1-1 ul" - u. " . - u .

i t (12) : o AX 1 At

It should be noted that Eq. ( 1 2 ) is not iterative in nature because the governing differential equa- tion [Eq. (11)) is linear. ?he second-order- accurate scheme given by Eq. (10) takes the form

n+l n = 0 ( 1 3 ) 2bX

I

For the purpose of convenience Eq. (13) uses a first-order-accurate, finite-difference in time Equations (12) and (13) demonstrate the simple fact that n + 1 grid points are required in order to develop a spatially nth order scheme that is fully upwind for the wave equation (in general the Euler equations would require 2n + 1 grid points in each spatial direction).

The problem with large stencils is that many points near the computational boundaries can n o longer be treated using the finite-differences used in the interior; therefore, they will require special treatment. Hence it is advantageous to develop finite-difference schemes that are as compact as possible subject to the constraints of stability, robustness, and other necessary and desirable properties. Fully upwind schemes have the disadvantage of requiring a large stencil for a given degree of accuracy. The natural question to ask at this point is whether one could combine compactness with the desirable properties of fully upwind schemes. ?he answer is decidedly yes; the obvious candidate being the upwind-biased scheme.

In Eq. (13) a three-point backward difference is used to represent the spatial derivative of u. This difference is second-order accurate. If we were to use the points i + 1 and i + 2 a l so in calculating ux, the finite-difference would become a fourth-order-accurate centered differ- ence. However, if only the point i + 1 is added to the existlng stencil, then the difference is upwind biased (more points to the left and fewer to the right) and can be made third-order accu- rate. The corresponding finite-difference repre- sentation of Eq. (11) is given by

"+1 "+ 1 ncl ncl u?" - un 2ui+l + 3ui - 6ui-l + u. 1-2 = 0

6Ax I + 1 At

(14)

5

In the case of the wave equation it seems as though higher-order accuracy is achieved at the expense of adding a grid point to tne stencil that i s required by the second-order scheme. However, this is not so in the more general case where signals propagate toward the left and right of a given point; both forward and backward differences are required to update the point in question. Third-order accuracy i c the general case is achieved by iising a grid poir,: to the right of point i in evaluating ti-e backiiard difference, and by using a grid point to t he left of point i in evaluating the forward differencc. An extension of this approach to the Euler and Navier-Stokes equations cac be found in Ref. 12.

The Fifth-Order-AcCUTate, tiiind-3iased Scheme

A fully upwind fifth-order scheme would require a stencil consisting of 1 1 points. How- ever, a fifth-orde~-acciirate, upwind-biased Scheme requires only 7 grid points. The ncmerical flux for Such a fifth-oPder-accurate, upwind-biased scheme is given by

Equation (15) is a simple extension of the second- order-accurate, numerical flux given by Eq. ( 9 ) and was obtained using simple Taylor Series expan- sions. Methods of constructing even higher-order- accurate fluxes are given in Ref. 16. rhe c o r r e - v sponding iterative-implicit scheme takes the form

i P I , ,+1/2 -.--I (16) A "

where r = p. order scheme, only the first-order part of thc numerical flux has been linearized in order to obtain the block-tridiagonal matrices on the left- hand side of Eq. (16). Also, the scheme given by Eq . (16) is only second-order-accurate in time.

A S i n the case of the second-

L Applying the fifth-order scheme to the simple wave equation [Eq. ( 1 1 ) l and using only first- derivatives in time we obtain

n + l n u . - " .

At 1

Using a simple Fourier analysis technique it c a n

stable. The leading dissipation term is a fiTth- order, sixth derivative of u instead of the third-order, fourth derivative found in Conven-

) tional central and upwind difference schemes. The fifth-order scheme as given by Eq. ( 1 6 ) requires approximately the same amount of computing tire as does the Second-order scheme. Detailed informa- tion regarding computing times for the second- order scheme can be found in Ref. 13.

) bc shown that Eq. ( 1 7 1 is unconditionally 1 120 + - [ J I A E - ( Q . ,+2, j ,'i+3, j"i+1/2, j

22AE-(Qi+~,j)Qi+2,j,ci+~/2,j

+ 12AE-(Qi .,Qi+l,j,<i+l/2,j ) , I

The fifth-order scheme has been developed + 6AE-(Qi-, . ,a. ) ] (15) thus far with the unsteady Euler equations in

, J 1 . J 91 mind. However, the scheme can be used, with few modifications, to solve the Navier-Stokes equa- tions. Consider the usnteady, thin-layer,

6

Navier-Stokes equations in two dimensions, and in the transformed coordinate system (r,,n,~)

v Or + zc + $ n = Re-'Sn (18)

where 6, E , and are as in Eq. ( 3 ) , Re is the Reynolds number, and the term is t.he viscous flux vector. The vector is given by

s =

where u is the viscosity, Pr is the Prandtl number, c the local speed of sound, and

K2 = p(qXu,, + n v ) / 3 Y n

Equations (18) and (19) assume that the body sur- face is a constant I I line (in incorporating the thin-layer approximation).

The fifth-order scheme as applied to Eq. (18) now takes the form

+ RtlS of E q . (16) (20)

where

ii = a 9 a O

and 6 is-a central-difference operator. The quantity Si,j+1/2 is the numerical flux corre- sponding to the viscous flux vector 9. This numerical flux can be calculated so that the numerical Scheme retains its fifth-order accu- racy. However, for this study, it was evaluated using Second-order-accurate, finite-differences; thus reducing the accuracy of the scheme (to second-order) in viscously dominated regions. This decrease in accuracy will not have any detri- mental effect on vortex preservation because the

1

viscous terms are negligibly small everywhere except-in the boundary layer and the wake. term Si,j+1/2 was evaluated using

The

where

1 1 ._ Qi,j+1/2 2 (Qi,j + Qi,j+l

Practical Aspects Regarding the Accuracy of the Scheme

In practice there are several factors that reduce the overall accuracy of the scheme. Some of these factors are listed below:

1) The accuracy of the surface and outer boundary conditions.

2) The Smoothness of the grid used for the computation.

3 ) The evaluation of the viscous terms

4) The accuracy of the patch-boundary condi- tion used to transfer information from patch to patch.

5 ) The accuracy of the schemes used at the points that are next to boundary points.

The conventional ways of treating the enti- ties mentioned above are usually consistent with the second-order accuracy of many of the schemes in current use. When these conventional ways are used with the fifth-order scheme developed in this study, the global accuracy of the scheme is reduced to second order. However, experience indicates that solutions that are thus obtained are far superior to those obtained with conven- tional Second-order Schemes (in spite of the drop in global accuracy). No attempt has been made in this study to improve the accuracy of the various boundary conditions used. Grid points that are neat to boundary points are treated using either central differences or a third-order-accurate upwind-biased scheme, depending on the size of the stencil available.

Vortex Preservation Tests

An understanding of the vortex-preserving property of a scheme can be obtained by simulating the flow associated with a vortex convecting in a free stream. A Lamb-type vortex is ideally suited to this calculation because the velocities remain finite at the core of the vortex. The pressure for such a vortex is a minimum at the center and increases asymptotically to the free-stream value as we move away from the center. A dissipative scheme is incapable of maintaining the minimum

value of the pressure (at the core of the vortex) at its original value; instead, the pressure at the center increases continually as the vortex convects with the flow. This change in pressure at the center of the vortex is one form of numeri- cally induced vortex decay. Therefore a good measure of the vortex preservine capability is the core pressure pCore.

type vortex convecting in a free stream is calcu- lated using a conventional Second-order-accurate, central-difference scheme (the Beam-Warming scheme) and the upwind-biased, fifth-order scheme. A portion of the grid used for the calcu- lation is shown in Fig. 1. It is a simple rec- tangular grid that is 55 core radii in length with equal spacing in the x- and y-directions (Ax = Ay). The complete grid extends outward a hundr'ed cope radii from the center in both the positive and the negative x- and y-directions. The exact solution corresponding to the vortex moving in a free stream is imposed on all the boundaries. This is possible since the exact velocity of the vortex with respect to the grid is known, and therefore, its position in the grid is ais0 known at all times.

In this test the flow associated with a Lamb-

Figure 2 shows the pressure contours at ini- tialization. At this point in time the center of the vortex is close to the left end of the grid shown in Fig. 1. The symbol at the center of the vortex seen in this figure (and in the following two figures) represents the analytical position of the vortex. Figure 3 shows the pressure contours obtained with the Beam-Warming scheme after 45 core radii of travel. The vortex is now close to the right end of the grid shown in Fig. 1. The grid spacing used for this calculation is A X = Ay = 1/3, the core radius of the vortex being 1.0. The distortion in the shape of the vortex near the cope is clearly seen. The second type of error that can be discerned in Fig. 3 is the dawn- ward motion of the vortex (the analytical center no longer corresponds to the center of the con- tours). Figure 4 shows the results obtained with the fifth-order scheme after 45 core radii of travel on the same grid. Clearly the distortions seen in Fig. 3 are absent in Fig. 4, thus indicat- ing the superiority of the new scheme in preserv- ing vortices.

Figure 5 Show the variation of the core Pressure pCore with the length of vortex travel. Curve A was obtained with the standard Beam-Warming scheme and first-order accuracy in time (and grid spacing values of AX = by = 1/3). I t is extremely dissipative and it is unsuitable for calculations requiring vortex Preservation. Curve B shows the amount of dissipation obtained on the Same grid with the Beam-Warming scheme and second-order accuracy in time. However, the scheme (for the grid density chosen) is still inadequate for vortex calculations. Curves A and B indicate the importance of

A considerable improvement is noticed,

second-order accuracy in time in performing unsteady vortex calculations. Curve C shows results obtained with the fifth-order-accurate scheme and the Same grid. There is a moderate amount of dissipation and the Scheme far this grid density barely makes it into the realm of vorter- blade interactions. Curves 0 and E W e w obtained with the central and fifth-order schemes, respectively, and on a grid with Ax = by : 1/11, Curves C and D have about the same amount of dissipation. Curve E shows almost negligiblc dissipation and indicates the grid dcnsity required for an accurate calculation with the fifth-order scheme.

w

The robustness of upwind-biased schemes, thc fact they do not need arbitrary smoothing parene- ters, and the low vortex decay rates of the f i f t h - order scheme led to the choice of this scheme for the vortex-blade calculation. This choice of scheme was also based on the other measures of distortion mentioned earlier.

Blade-Vortex Interaction

Having chosen an appropriate algorithm, we nou proceed to calculating blade-vortex interac- tions. In keeping with the experiments of Ref'. 3, the airfoil used in all of the following calcula- tions was a NACA-0012 airfoil. The calculations Were performed using the unsteady, thin-layer, Navier-Stokes equations. The Baldwin-Lornan

and Sutherland's Law was used to determine the natural viscosity of the fluid. The other impor- tant aspects of the calculation such as the bound- ary conditions, the initialization procedures, and the grids used are discussed below.

Grid System for the Blade-Vortex Calculation

model17 was used to calculate the eddy viscosity L'

As the vortex preservation tests of the prc- vious section indicate, a minimum grid density is required for accurate calculations even with the fifth-order scheme. An approximate value far this grid density is about four grid cells per c o r e radius of the vortex. For the relatively small vortices considered in this study (about 1 in. i n core radius), a grid with a minimum grid density of four cells per core radius would require an enormous number of grid points. However, Such large grid point densities are only required in the path of the vortex and in the boundary layer and wake associated with the airfoil (the grid densities in the boundary layer and in the wake are actually several orders of magnitude larger). The simplest approach to selectively refining the grid, and thus minimizing the number of grid points required to perform the calculation, is to use the patched-grid method of Refs. 7-9.

Figure 6 shows the three zones used to repre- sent the region of interest. The region as a v whole is a "C" type of region Surrounding the airfoil, the outer boundaries being approximately

a

10 chords away in every direction from the center of the airfoil, Zones 1 and 2 are symmetric. Zone 3 is a tubelike region that extends from the airfoil all the way to the outer boundary of the region, place only in zone 3; hence only zone 3 contains the high grid densities required to preserve the vortices. Figure I shows a representative grid for zone 3. zones in the vicinity of the airfoil. The grids in all the zones were generated using a combina- tion of algebraic and elliptic (Ref. 18) grid generation procedures. The grids are orthogonal to the surface of the airfoil (this feature cannot be seen in Fig, 8 because of the closely spaced grid points in the boundary layer). Zones 1 and 2 are discretized with 121 x 151 grids and zone 3 is discretized with a 91 x 311 grid, a total of approximately 65,000 grid points.

Boundary Conditions

L Vortex travel ahead of the airfoil takes

Figure 8 shows the grids in all three

The use of multiple grids in simulating the flow over the airfoil configuration shown in Fig. 6 results in several computational bound- aries. these boundaries is briefly outlined below.

The boundary conditions used at each Of

The lower boundaries of a l l three grids ( n = 0) correspond to the airfoil surface; hence, the "no slip" condition and an adiabatic wall condition are imposed at these boundaries. addition to the no slip condition, the derivative of pressure normal to the wall surface is Set to zero. The pressure derivative condition, the adiabatic wall condition, and the equation Of state tocether yield

In

_ _

ae u u v @ an - p an + ii an

where n is the direction normal to the airfoil surface. These boundary conditions are imple- mented in an implicit manner by using the follow- ing equation instead of Eq. (20) to update the grid points an the airfoil surface.

where

C:

L ..

and

Ji 2 a = - - 1.1

J .

"wall 'wall

6 : - -

Equation (21) is an implicit, spatially first- order-accurate implementation of the no slip adia- batic wall condition (first-order-accurate because the zero normal derivative condition is imple- mented using a two-point forward difference). second-order-accurate, three-point, forward- difference corrector step is also implemented after each time-step. It should be noted that Eq. (21) requires the grid to be orthogonal at the airfoil surface and the Jacobians of the transfor- mation J - and 3 . to be independent of T.

A

1,1 1,2

The upper boundaries of all three zones (n z n a x ) are subsonic inlet boundaries. quantiTies need to be specified at these bound- aries. The three chosen for this study are the generalized Riemann invariants

Three

2c r - 1

nt + uIlx + vn R 1 = -~

R -

R - e 3 - p Y

The dependent variables p , u , v, and p in Eq. (22) are taken to be the free-stream values under the following conditions: (1) the airfoil is at zero angle of attack and (2 ) the grid does not contain a vortex. At a Finite angle of attack the values of p , u, v, and p are chosen so that the value of the circulation on the outer boundary is consistent with the lift of the airfoil at the given angle of attack. The importance of this modification to the boundary condition is dis- cussed in Ref. 19. If the grid system contains a vortex then the dependent variables are taken from the composite "vortex in a free-stream" solution (the values of P , u, v, and p used in Eq. (22) change with time in this case because of the motion of the vortex). is necessary to update the points on this bound- ary) is also a Riemann invariant

The fourth quantity (which

9

and is extrapolated from the interiors of zones. The manner in which these boundary conditions can be implemented implicitly is described in Refs. 20 and 21.

The right boundary of zone 1 and the left boundary of zone 2 are subsonic exit boundaries. A simple, implicit-extrapolation procedure fol- lowed by a n explicit, post-update correction is used at these boundaries. The implicit part of the boundary condition for zone 1 is implemented as follows:

This step is followed by thc explicit correction

Once again, the free-stream pressure p, is replaced by the pressure corresponding to the composite vortex in a free-stream solution if the grid system contains a vortex. The exit boundary of zone 2 is treated similarly.

The equations necessary to transfer informa- tion in a time-accurate manner across the patch- boundaries separating zones 1 and 3, and zones 2 and 3 have already been developed in Refs. 7-9. The development of these equations is fairly involved; therefore, in the interest of brevity, it is not included here. The only remaining boundary is the wake boundary separating zones 1 and 2. Although the grid lines of zones 1 and 2 are continuous across the wake, for the purpose of convenience the wake boundary is also treated as a patch boundary. condition can be found in Refs. 7-9 .

Initialization Procedures

Uetails of the patch-boundary

As indicated earlier, a Lamb-type vortex is ideally Suited for the calculation because of the finite velocities at the core of such a vortex. Also, the vortex used in the experimental investi- gation of Ref. 3 has a structure that resembles a Lamb-type vortex. structure of the vortex used in the calculation to closely resemble the experimental one, the cope radius, the strength of the vortex, and the value O f the peak velocity must all be identical in the experimental and the analytical vortices. erence 11 uses an analytical vortex structure whose constants are determined from several

In order for the analytical

Ref-

experiments. The cylindrical velocity cornponcnt v B for this vortex is g iven by

where a is the core radius of the vortex, I_

represents the strength Of the vortex, r is the distance from the center of the vortex, and u _ is the free-stream velocity whose numerical value depends on the type of nondimensionalization used. The vortex used in Refs. 3 and 4 corre- sponds to r = -1.477 and a = 0.9811 in. The negative value of r results in a clockwisc floW direction ( f o r the vortex in isolation).

In this study, the code is validated for the vortex-blade interaction case using the vortex given in Eq. (25). The pressure and density fos' the vortex flow are obtained as

2 P V

dr r !QB (261

where the density o is Obtained from

Equation (26) is integrated in conjunction with Eq. (27) using a Runge-Kutta scheme. The differ- <-' entia1 equation is integrated from a large value of r (where the pressure and density are known) inward to the center of the vortex. Equa- tions (25)-(27) represent a stationary vortex. It can be easily shown that this flow field exactly satisfies the steady Euler equations. The effect of the free stream is brought about by adding the Cartesian velocity components of the free stream to the Cartesian velocity components of the 'vortex described by Eqs. (25)-(27).

We now discuss the calculation procedure tused to study vortex-blade interactions. Initially the grid points of all the grids are set to free- stream values. Equation (20) with the appropriate boundary conditions is then integrated to coever- gence on the grid system presented earlier for t.he airfoil (airfoil without a vortex). Finally the vortex is initialized at a distance of apppoxi- mately 7 chord lengths ahead of the airfoil. The convection of the vortex with the fluid and the subsequent interaction of the vortex with the airfoil is then monitored.

Results

The three-zone, unsteady, thin-layer Navic-r-

v Stokes code was first validated by calculating the flow over a NACA-0012 airfoil. The free-stream Mach number was chosen to be 0 . 8 , the Reynolds number was 1.0 x 10 6 based on the chord, and the

angle of attack was 0". Figure 9 shows the pres- sure contours obtained at convergence. The smooth transition of the contour lines from zone to zone at the interface boundary is clearly seen. I n

ious oscillations, the accuracy of the method was dropped to second-order in the vicinity of the shock, and flux limiters of the type developed in Ref. 10 were employed in this region. The shock- capturing quality of this procedure is evident from Fig. 9. Figure 10 shows a comparison of the surface pressure coefficients obtained with the present scheme and with the ARC2D code (private communication, Dr. T. H. Pulliam, NASA Ames Research Center). The two sets of results are in good agreement.

The second validation study consisted of

- order to capture the shock without obtaining spur-

simulating a two-dimensional analog of the experi- ment of Ref. 3 . A schematic of the configuration (reproduced from Ref. 4) used in Ref. 3 is shown in Fig. 1 1 . It consists of a two-bladed ro tor in a uniform free stream. The angle $ measures the azimuthal position of the blade. The rotor blade interacts with the tip vortex generated by a NACA-0015 airfoil that is positioned upstream of the rotor. At approximately * = 180° the test section A-A of the rotor undergoes an interaction with the vortex commonly referred to as parallel blade-vortex interaction. The flow at this sta- tion can be roughly approximated as a two- dimensional, blade-vortex interaction. Further details regarding this approximation can be found in Ref. 4.

In order to perform the second validation calculation the flow field was initialized with a free-stream Mach number of 0.536 (this corresponds to a rotor tip Mach number of 0.6). The vortex was then released at approximately 7 chord lengths from the leading edge of the airfoil and 2.4 in. below the axis of symmetry (yv = -2.4). The vortex strength used for this simulation was r = -1.477 and the core radius was 0.984 in. The Reynolds number used in the calculation was 1.3 x lo6 based on the chord, the chord length of the airfoil being 6 in. The free-stream tempera- ture was taken to be 530" R. Figure 12 shows the variation of the lift coefficient as a function of the vortex position. The lift is initially nega- tive because the vortex induces a downwash at the leading edge of the airfoil. As the vortex passes the leading edge of the airfoil, the lift coeffi- cient rapidly increases to a positive value. This change in sign in the lift coefficient occurs because the vortex begins to produce an upwash instead of a downwash. The lift on the airfoil should eventually vanish as the vortex travels far downstream of the airfoil. However, this final portion of the interaction was not computed.

Figure l3a shows the surface pressure dis- tribution at the rotor azimuthal position * = 182.65". This azimuthal position corresponds to xv 1.74 in., where xv represents the dis- tance from the leading edge of the airfoil to the

I

center of the vortex (measured as positive in the positive x-direction). The full and dashed lines represent the present results, and the symbols depict the experimental data of Ref. 3. The com- parison between experiment and theory is fairly good. The area between the two curves represents the negative lift the airfoil experiences at this azimuthal position. Figure 136 depicts the sur- face pressure distribution at $ = 185.95O (xv I 3.9 in,). The comparison between theory and experiment is quite good on the lower side, but it is not as good on the upper side of the airfoil. The lift on the airfoil has changed from negative to a small positive value at this azimuthal pasi- tion (the numerically obtained transition point for the lift being 185O). Figure 13c shows the pressure distribution at # = 192.85" (xv = 8.4 in.). experiment and theory is much better on the lower side of the airfoil than it is on the upper side.

Once again the comparison between

I n comparing the experimental results of Ref. 3 with the numerical ones of the current effort, it must be remembered that the two- dimensional flow field being numerically simulated here is only an approximation to the real three- dimensional flow field of Ref. 3. In the light of this approximation, the comparison between theory and experiment, as seen in Figs. 13a-l3c, can be considered to be quite good. To further validate the new algorithm and code, the numerically obtained surface pressure distributions of Figs. 13a-13c are compared with the numerical results of Ref. 4 in Figs. 14a-lhc. In Figs. 14a-14c the symbols represent the numerical data of Ref. 4 . The comparison between the two sets of results is seen to be good. The perturba- tion approach of Ref. 4 is quite adequate for this case because of the relatively weak interaction between the airfoil and the vortex. However, it must be remembered that the present approach does not require any vortex-sustaining mechanism such as the perturbation approach of Ref. 4 which requires knowledge of the shape, size, and trajec- tory of the vortex in order to be effective; instead, the current approach uses only a straightforward finite-difference scheme to calcu- late the interaction.

Having validated the code, a strong interac- tion case where the vortex collides "head-on" with the airfoil was calculated'. The free-stream con- ditions for this case were identical to those used in the previous case. The vortex strength r was increased in magnitude to -2.382, and the core radius was taken to be 0.984 in. The vortex was once again released about 7 chord lengths ahead of the airfoil, but at yv = 0.0 instead of at yv = -2.4 as in the previous case. Fig- ures 15a-15d depict vorticity contours at the azimuthal positions $ = 175O, 180", 185", and 195". to show the interaction lies in the fact that this approach results in two distinct entities: the vortex and the airfoil with its boundary layer and wake. In Fig. 15a the vortex is seen approaching

The advantage of using vorticity contours

the airfoil. The vorticity contours associated with the vortex show little deviation from their original concentric circular shapes. I n Fig. 15b the vortex is seen to be colliding with the air- foil and splitting into two separate vortices. Figure 15c shows two vortices, one above the air- foil and one below the airfoil. The lower vortex has moved a greater distance to the right because of the larger convecting velocities on this side. Figure 15d shows the interaction of the vortex with the wake of the airfoil. A recambina- tion of the two vortices is not evident. The boundary layer on the airfoil was found Lo be attached through this strong vortex-blade inter- action. It is possible that turbulence plays a large ro le in suppressing leading-edge separation and consequent secondary vortex generation. R laminar calculation of the head-on encounter is currently under way. Extensive separation and generation of secondary vortices in the laminar case w i l l indicntc the necessity for improved turbulence modeling (for the turbulent calculation).

The zo11al boundaries separating 20nes 1 and 3, and Zones 2 and 3 are seen in Figs. 15n-15d. Some of the Vorticity c o n t o u r s a r e discontinuous across these boundaries. These discontinuities appear because only the dependent variables are forced to be continuous across zonal boundaries; no control is exercised on the deriva- tives of the dependent variables. Pressure con- tours that are shown later in this section will not exhibit any discontinuities at the zonal boundaries.

Figures 16a-16d show pressure contours at the same azimuthal positions as those at which the vorticity contours were displayed in Figs. 15a-15d. Figure 16a shows the vortex approaching the airfoil. Unlike the vorticity contours at this position, the pressure field of the vortex is seen to be distorted from its origi- n a l circular nature. In Fig. 16b the pressure field of the Vortex has completely merged with the pressure field of the airfoil. The distorted circles of the previous figure are absent in this figure. Figure 16c shows the acoustic disturbance (associated with tho blade-vortel interaction) beginning to form and to propagate outward. Fig- ure 16d shows the remnants of the original vortex pressure field in thc wake region of the air- foil. field corresponding to Fig. 16d. The acoustic Wave is clearly Seen in this figure. Fig- ures 18a-18d correspond to Figs. 16a-16d and are magnified views of the leading edge showing the movement of t h e stagnation point as the lift changes from negative to positive.

Figure 17 shows a larger view of the flow

Figures 15a-15d and 16a-16d show the tremen- dous modification that the original pressure and vorticity fields of the vortex undergo during the interaction process. Because of this reason, any method that relies on a priori information regard- ing the shape, size, and trajectory of the vortex

to calculate the interaction effects will be totally inadequate. The fifth-order scheme does not require any such information. Herein lies the efficacy of the present approach. 'W

The second strong interaction case that was studied consisted of 'transonic flow past the air'. foil. The free-stream Mach number chosen for this calculation was 0 . 8 . The Reynolds number, f r e e - stream temperature, and vortex structure werc the same as in the previous case. The flow field associated with the airfoil i n isolation Was first calculated, and the vortex was then released a h w t 7 chord lengths upstream of the leading edgc at y v = -1.56 in. contours a t selected azimuthal positions. Fig- ure l9a shows the vortex approaching. The contour lines of the vortex are as yet undistorted. The asymmetry of the flow field can be discerned from the asymmetry of the upper and lower airfoil shocks. In Fig. 19b the vortex is almost at the Same axial position as the leading edge of the airfoil. An appreciable change in vortex shape has occurred at this Station. Figure 19c shows the interaction between the lower shock and the vortex. The boundary layer was found to thicl<en considerably in the vicinity of the shock, and a large bubble O F separation was found to extend from the foot of the shock to the trailing edge of the airfoil. Figure 19d shows the vortex aftcr it had passed through the shock. Surprisingly, it seems to have regained most of its original circu- lar nature.

Figures 19a-19d show vorticity

i-.

Figures 2Oa-2Od display pressure contours for the same case. The approach of the vortex is seen in Fig. 20a. In Fig. 20b the vortes pressure field has merged completely with the airfoil pres- sure field, Figure 20c shows the vortex passing through the shock. This passage of the vortex results in substantial changes in the lower shock structure; this shock was observed to bifurcate at the airfoil surface. Figure 20d shows the lower shock returning to its original configuration. Some of the asymmetry of the flow field has disap- peared and the vortex is seen about 1 chord length downstream of the airfoil. Figure 21 shoua pres- sure contours at an intermediate position (izortex passing through the shock), and is included h e m to show the bifurcation of the lower shock a t both the upper and lower e n d s . The bifurcation at the lower end for a weak interaction Case has beer, reported in Ref. 4 .

Conclusions

A major problem in performing vortex calcu- lations with conventional finite-difference schemes is the rapid decay of the vortex structure that occurs because of numerical dissipation. This study presents a fifth-order-accurate, upwind-biased scheme, based on the Osher-type of flux differencing, that preserves vortex structure for much longer times than existing Second-Order-accuPate, central and upwind

difference schemes. Vortex calculations demonstrating this aspect of the fifth-order scheme are also presented.

u A thin-layer, Navier-Stokes code for vortex- blade CalculationS has been developed. The code employs the new fifth-order scheme to preserve vortices ovcr long periods of travel. The code also has a multizone capability to selectively refine the grid only in the path of the vortex so that grid point requirements are minimized. The code has been validated for a weak vortex-blade interaction case using both experimental data and earlier numerical data. The new scheme made unnecessary the use of vortex preserving tech- niques such as perturbation methods (which require a priori specification of vortex shape, size, and trajectory during the interaction process).

The strength of the new code lies in the fact that it can be routinely used for strong interac- tion cases without any modification (thls is once again because the code does not require any infor- mation regarding the vortex). This capability was fully exercised by calculating the case of a direct collision between a vortex and an airfoil at subsonic speeds. 'The vortex was found to split into two vortices. The resulting complex interac- tions are documented in the paper. A second application of the code was a vortex-blade inter- action at transonic speeds. The deformation of the vortex as it passes through the airfoil shock, the changes in the structure of the shock during ., the same process, and the effect of the interac- tion on the airfoil boundary layer are a l l discussed.

Whereas the applications of the code to the strong interaction cases have yielded some i.nter- esting fluid physics, hopefully the main contribu- tion of this paper lies in the computational philosophy that it professes, that is, an approach wherein the degree of modeling of the physics is minimized even if the computational costs are increased. This approach will become i.mpor'tant in Sclvine large scale problems, such as the heli- copter rotor, i n a more Straightforward fashion.

Acknowlede_ments

I would like to thank Dr. G. R . Srinivasan of JAI Associates and Dr. W. J. McCroskey of the U.S. Army Aviation Research and Technology Activity Branch for the many stimulating discussions I haYe had with them regarding blade-vortex interactions and for the encouragement that they have provided during this investigation.

References

'Caradonna, F. X., Desopper, A., and Tung, C., "Finite Difference Modeling of Rotor Flows Including Wake Effects," Paper No. 2.7, Eighth European Rotorcraft Forum, Aix-en-Provence, France, Sept. 1982.

'Straw", R. C. and Tung, C., "The Prediction of Transonic Loading on Advancing Helicopter Rotors," AGARD/FDP Symposium on Applications of Computational Fluid Dynamics in Aeronautics, Aix- en-Provence, France, April 1986.

3Caradonna, F. X., Laub, G. H., and Tung, C., "An Experimental Investigation of the Parallel Blade-Vortex Interaction," 10th European Rotor- craft Forum, The Hague, Netherlands, Rug. 1984.

4Srinivasan, G . R . , McCroskey, W. J . , and Baeder, J. D . , "Aerodynamics of Two-Dimensional Blade-Vortex Interaction," A I A A Paper 85.1560, AIAA 18th Fluid Dynamics and Plasmadynamics and Lasers Conference, Cincinnati, Ohio, July 16-18, 1985.

5Beam, R. M. and Warming, R. E . , "An Implicit Factored Scheme for the Compressible Navier-Stokes Equations," Proceedings of the AIAA 3rd Camputa- tional Fluid Dynamics Conference, Albuquerque, New Mexico, June 27-28, 1971.

6Sankar, N. L. and Tang, W., "Numerical Solu- tion of Unsteady Viscous Flow Past Rotor Sec- tions," AIAA Paper 85-0129, AIAA 23rd Aerospace Sciences Meeting, Reno, Nevada, Jan. 14-17, 1985.

7Rai, M. M . , "A Conservative Treatment of Zonal Boundaries for Euler Equation Calculations," Journal of Computational Physics, Vol. 62, No. 2, Feb. 1986, pp. 472-503.

'Rai, M. M . , "An Implicit Conservative Zonal Boundary Scheme for Euler Equation Calculations," AIAA Paper 85-0488, AIAA Z3rd Aerospace Sciences Meeting, Reno, Nevada, Jan . 14-17. 1985.

9Rai, M. M., "Navier-Stokes'Simulations of Rotor-Stator Interaction Using Patched and Over- laid Grids," AIAA Paper 85-1519, AIAA 7th Computa- tional Fluid Dynamics Conferencc, Cincinnati, Ohio, July 15-17, 1985.

"Chakravarthy, S . R. and Osher, S., "High Resollition Applications of the Osher Upuind Scheme for the Euler Equations," RIAA Paper 83-1943, AIAA 6th Computational Fluid Dynamics Conference, Danvers, Massachusetts, July 13-15, 1983.

"Roe, P. L., "Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes," Journal of Computational Physics, Vol. 43, 1981, PP. 357-372.

13

12Chakravarthy, S . R., Szerna, K . - Y . , Goldberg, U. C., and Gorski, J. J., "Application o f a New Class of High Accuracy TVD Schemes t o the Navier-Stokes Equat.ions," AIAA Paper 85-0165, A I A A 23rd Rerospace Sciences Meeting, Reno, Nevada, Jan . 14-17, 1985.

13Rai, M. M. and Chakravarthy, S. R . , "An Implicit Form for the Osher Upwind Scheme,'' AIAA Journal, Vol. 24, No. 5, May 1986, pp. 735-743.

14Chakravarthy, S . R . , "Implicit Upwind Schemes Without Approximate Factorization," AIAA Paper 84.0165, AIAA 22nd Aerospace Sciences Meet- ing, Reno, Nevada, Jan. 9-i2, 1984.

I5Rai, M. M., "A Relaxation Approach to Patched-Grid Calculations with the Euler Equa- tions," Journal of Computational Physics, Val. 66, No. 1 , Sept. 1986, pp. 99-131.

"Osher, S . and Chakravarthy, S . R., "Very High Order Accurate TVD Schemes," ICASE Report No. 84-44, Sept. 1984.

17Baldwin, B. S. and Lomax, H., "Thin Layer Approximation and Algebraic Model for Separated Turbulent Flow," RIA& Paper 78-251, RIA& 16th Aerospace Sciences Meeting, Huntsville, Alabama, Jan. 16-18, 1978.

Fig. 1 Grid for vortex preservation tests.

"Steger, J . L. and Sarenson, R . L., "Auto- matic Mesh-Point Clustering Near a Boundary in Grid Generation with Elliptic Partiai Differential Equations,'' J, Ccmp. Phys., Val. 33, No. 3, Dec.

19Pulliam, T. H. and Steger, J . L., "Recent

1979, pp. 405-410. W

Improvements in Efficiency, Accuracy, and Conver- gence fo r Implicit Approximate Factorization Algo- rithms," AIAA Paper 85-0360, AIAA 23rd Aerospace Sciences Meeting, Reno, Nevada, Jan. 14-17, 1984.

"Chakravarthy, S. R . , "Euler Equations-- Implicit Schemes and Boundary Conditions," AI A A Journal, Vol. 21, NO. 5, May 1983, pp. 699-706.

*'hi, M. M. and Chaussee, D. S . , "New Implicit Boundary Procedures: tion," AIAA Journal, Vol. 22, No. 8, Rug. 1984, pp. 1094-1100.

Theory and Applica-

Fig. 2 Vortex pressure contours at initialization.

14

Fig. 3 Vortex pressure contours after 45 core radii of travel (second-order-accurate central- difference scheme).

Fig. 4 Vortex pressure contours after '15 core radii of travel (fifth-order-accurate upwind- biased scheme).

A --&- CENTRAL DIFFERENCE SCHEME,

6 --C-- CENTRAL DIFFERENCE SCHEME (Ax = Ay = 1/31 D -U- CENTRAL DIFFERENCE SCHEME ( A x = AY = 1/41 C U UPWIND SCHEME IAx = Ay = 1/31 E -0- UPWIND SCHEME (Ax = Ay = 114)

FIRST ORDER IN TIME (Ax = Ay = 1/31

.88 r 87

86 8

E

P 1s 85

a

84

.82 I I I I I 0 5 10 15 20 25 30 35 40 45

NUMBER OF CORE RADII TRAVELLED

Fig. 5 Vortex decay rates for various schemes.

15

ZONE 1

ZONE 2 \

Fig. 6 Zoning and outer boundaries used for thm vortex-blade calculation.

Fig. 7 Representative grid for zone 3.

Fig. 8 Grids for all three zones in the vicinity of the airfoil,

I-..

Fig. 9 Pressure contours at convergence For the airfoil in isolation.

1 6

-1.0

-.75

-.50

-.25

0

1 .25 0 P

50

.75

1.0

1.25

- FIFTH-ORDER SCHEME 0 ARC2D

-.2 0 .2 .4 .6 .8 1 .o x l c

Fig. 10 Surface pressure distribution for the airfoil in isolation.

$ fl NACA 0015 WING

-

F i g . 1 1 Schematic of the experimental configura- tion of Ref. 3.

t 2 0

160' 170" 180 190c 200" - 2 'L

160 170" 180 190c 200" + Fig. 12 Variation of the airfoil lift coefficient with rotor azimuthal position

17

NUMERICAL RESULTS - AIRFOIL LOWER SIDE __.. AIRFOIL UPPER SIDE EXPERIMENTAL RESULTS

8 AIRFOIL UPPER SIDE -.75 0 AIRFOIL LOWER SIDE

P 0

i; = 182.65

.25

1 a

0

2 5

-,75 l a

-.50

0

.25 0 2 5 50 .75 1.0

xlc

Fig. 13 Experimental and numerical surface pres- sure distributions at various azimuthal angles. a) * = 182.65O; b) p = 185.95"; c) $ = 192.85".

PRESENT RESULTS __ AIRFOIL LOWERSIDE .~~..-. AIRFOIL UPPER SIDE RESULTS OF REF. 4

0 AIRFOIL LOWER SIDE a AIRFOIL UPPER SIDE

W

-.75 1

1

0 .25 50 .75 1.0 x/c

Fig. 14 A comparison of surface pressure distri- butions obtained with the current scheme and the perturbation scheme of Ref. 4. a) V = 182.65"; b) $ = 185.95"; c) $ = 192.85".

18

\

ii 180'

\

Ibl

185'

Id1

fig. 15 Vorticity contours at va r ious azimuthal ang1.e~ for the case y, = 0.0 i n . n_ = 0.536. a ) v = 175.00"; b ) B = 180.00 ' ; c) b = 185.00" ; d ) v z 195.00°.

and

1 9

- $ = 175 / / \ 185"

Fig. 16 Pressure contours at various azimuthal angles for the case yv = 0.0 in. and M_ I 0,536. a) p c 175.00'; b ) $ = 180.OOo; c ) = 185.00"; d ) P = 195.00°.

V

20

v

Fig. 11 Pressure contours at * = 195.00' showing the propagation of the acoustic wave.

21

180" \

CJ

d i

Fig. 18 yy = 0.0 in.

Pressure contours near the leading edge at various azimuthal angles For the case and M _ = 0.536, a) 175.00°; b ) $ = 180.00°; c ) 0 = 185.OOa; d) 0 = 195.00"

22

+5 = 175” \

.. 180‘

\ \ \

b)

190”

i1

Fig. 19 Vorticity C O ~ L O U P S at var ious a7.imuthal angles fo r the case y, = -1.56 in. and Me, 0.8. a ) # i 175.00”; b) 6 = 180.00”; c ) * i 185.00’ ; d ) $ I 195.00°.

23

\ I I 180"

'APPROXIMATE AXIAL LOCATION OF VORTEX

APPROXIMATE AXIAL LOCATION CJ OF VORTEX

-__

F i g . 20 Pressure contours a t var ious azimuthal angles for the case y, = -1.56 i n . and M, = 0.8. a ) V = 175.00'; b) V 2 180.00"; c) @ z 185.00"; d ) * : 195.00O.

24

W

Fig. 21 Pressure

--

contours at 0 = 188.00" showing shock bifurcation caused by shock-vortex interaction.

25