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RD-RI59 123 PSEUDOSPECTRAL CALCULATIONS OF TWO-DIMENSIONAL 1/1TRANSONIC FLOW (TRSK 1) NU..(U) FLOW INDUSTRIES INCKENT WA W H JOU ET AL. SEP 84 TN-215 RFOSR-TR-84-ii74
UNCLSSIFIED F49620-82-C-0022 F/G 29/4 N
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Progress Report for
o Contract No. F49620-82-C-0022Lfl
< TASK 1. PSEUDOSPECTRAL CALCULATIONS OF TWO-DIMENSIONAL TRANSONIC FLOW
TASK 2. NUMERICAL INVESTIGATION OF VTOLAERODYNAMICS
September 1984
Submitted to
Air Force Office of Scientific Research
TDTICELECTENMJAN 2 3 55
Flow Industries, Inc.
Research and Technology Division- B21414-68th Avenue SouthKent, Washington 98032
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AFOSR-TR" <;4,. 1746. NAME OF PERFORMING ORGANIZATION b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION
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11 TITLE (Include Security Claaification) TASK 1. FSEUDOSPECT CALCULATIONS OF TWO-D IMENSIONAL tRANSONIC
FLOW TASK 2. NUMERICAL INVESTIGATION OF VTOL AERODYNAMICS (UNCLASSIFIED)12. PERSONAL AUTHORS)•
JOU, WEN-HUEI: METCALFE, R W13a. TYPE OF REPORT 13;. TIME COVERED 14. DATE OF REPORT (Yr. Mo.. Day) 15. PAGE COUNT
FINAL FROM 9FrR1 TO2lInEfC 1984, September 5416. SUPPLEMENTARY NOTATION
17. COSATI CODES 18. SUBJECT TERMS (Continue on reverge if neceusary and identify by block number)
FIELD GROUP SUB GR. TRANSONIC FLOW ")NAVIER-§TOKES SOLUTIONS>IPECTRAL METHODS JET-GPOUND INTERFERENCE.JET IMPINGING FLOWS.
IS. ABSTRACT (Continue on reverse if necesary and identify by block number "
The status of the research performed unde AFOSR contract F49620-82-C-0022,is reported intwo FLow Research Technical Notes& "Pseudospectral Calculations of Two-Dimensional Trans-onic Flow' and "A Numerical Inyestigation of VTOL Aerodynamics." The formulation of theproblem, the method of solution, and numerically obtained results arepresented for eachcase. .
20. DISTRIBUTION/AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION
UNCLASSIFIED/UNLIMITED 9 SAME AS RPT. C3 DTIC USERS E- UNCLASSIFIED
22s. NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE NUMBER 22c OFFICE SYMBOL
Dr James D Wilson (Include Arco Code)202/767-4935 AFOSR/NA
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-•. I1.>.X>. - ;..... . . .
Foreword
0The year-end progress on Contract No. F49620-82-C-0022 for the Air Force
Office of Scientific Research is reported in this document. The work for.
Task 1 is described in "Pseudospectral Calculations of Two-iesoa
Transonic Flow," Technical Note No. 215, which is provided first. A
description of the Task 2 work, "A Numerical Investigation of VTOL
Aerodynamics," Technical Note No. 216,
is included next.
Accession For 4
NTis -GRA&I I ASCDTIC TABUnnnc~neo' AI 10I- C)1
By_. a---'.
AvPI>Qt~iity Codes jnrcrw,1 tlO
- AvaiI1 and/or Cif e~~)cDisat Special
4I
5
. . . .. N .. . . . .
Flow Technical Note No. 215
PSEUDOSPECTRAL CALCULATIONS OF
TWO-DIM4ENSIONAL TRANSONIC FLOW*
by
Wen-Huei iou
R. W. Metcalfe
September 1984
*This work is supported by the Air Force Office of Scientific Research and theOffice of Naval Research under Contract No. F49620-82-C-0022.
Table of Contents
Page
List of Figures ii
1. Introduction 1
2. Basic Approach 2 .
2.1 Governing Equations 2
2.2 Numerical Scheme 2
2.3 Filtering 3
2.4 Boundary Conditions 4
2.5 Convergence Acceleration 5
3. Computed Results for Hybrid Scheme 7
4. Full Spectral Scheme 9
5. Conclusions 11 ..
References 12
Figures 13
Appendix: Pseudospectral Calculations of Two-Dimensional
Transonic Flow 17
TN-215/09-84
- .o . •.
*. . . . .. . . . . . . . . . . . . .
.'.* % *.*..* **.-*%-*.*-iS. * * . .. . . . . . ~ **. .- . .. ,.
List of Figures
Page
Figure 1. Pressure Distribution for a Subcritical Flow Overa Cylinder at Mach 0.39 13
Figure 2. Pressure Distribution for a Supercritical Flow Overa Cylinder at Mach 0.45 14
Figure 3. Pressure Distribution on a Karuian-Trefftz Airfoil atMach 0.75 Using a 64x24 Grid (350 Steps) 15
Figure 4. Comparison of Results with a Finite Volume Scheme fora Karman-Trefftz Airfoil at Mach 0.75 16
TN-21/09-8
1. Introduction
In recent years, there has been strong interest in applying pseudospectral
methods to various flow problems. Some examples of these applications are
collected in a recently published book by Voight, Gottlieb, and Hussaini
(1984). One of the current areas of interest is the calculation of compres-
sible flows with shock waves. Gottlieb, Lustman, and Orszag (1981) have
investigated the one-dimensional shock tube problem using a pseudospectral
method and reported the ability of capturing the shock within one grid point.
Gottlieb, Lustman, and Streett (1982) have reported work on two problems in
this area. The first is the solution of Euler equations for oblique shock
reflection from a flat plate. By using a sparse 8x8 computational grid, they
have shown that it is possible to capture the shock wave within one grid point
and that the treatment of boundary conditions is extremely crucial to the con-
struction of a stable scheme. The second problem is the solution of a full
potential equation for transonic flow past an airfoil. An airfoil is mapped - .
to a circle. In the circumferential direction, a Fourier series is used, while
Chebyshev expansion is used in the radial direction. In-order to stabilize
the calculation, an artificial viscosity is used in the governing equation.
Gottlieb et al. (1982) has shown that in a subsonic flow, a highly accurate 0
solution can be obtained by using a sparse grid of 32x8 grid points. In the
transonic case with shocks, the shock wave spans across three grid points and,
hence, is not accurate enough if a sparse grid is used. The problem chosen by
Gottlieb et al. (1982) for the solution of the Euler equations is, unfor-
tunately, not a good one. The region between the oblique shock is a constant
state. Aside from capturing the shock wave, their research yields no informa-
tion about the accuracy of the method in the region of smooth variation away
from the shock wave. Their work on potential flows has shown that the .
incorporation of conventional artificial viscosity into the spectral method
will cause deterioration of the accuracy and thus defeat the purpose of using -
a spectral method.
In the present work, we have chosen the realistic problem of transonic
flow over an airfoil to study the application of a spectral method to compres-
sible flows with shock waves. Part of the findings of the present work have
been reported in a conference paper (Jou, Jameson, and Metcalfe, 1983), which
is included in this report as an appendix. -
TN-21 5/09-84 *..-*.-...-$
7S
-2-
2. Basic Approach
2.1 Governing Equations
The basic approach is to map the exterior of an airfoil to the interior of
a circle. Polar coordinates in the mapped plane will be used. Spectral
decomposition of the solution can be used in the mapped plane. To serve this
purpose, the Euler equations in the mapped plane are written in fully
conservative form by using both physical and contravariant velocities. These
equations are
q + (T) + a y = 0 (1)
where
2 21Pv u +v yP/yM ; o = yPT
pvU - X.YP/YM D PvV + xXP/YM O
LI LHU L PHV
[U] yy -x [] ; H f [Cy - l)P + E]/pYxX x -, --'
(x,y) are Cartesian coordinates in the physical plane, (X,Y) are polar coor-
dinates in the mapped plane, P is the density, (u,v) are velocity components,
(U,V) are unscaled contravariant velocity components, E is the energy, H is
the specific enthalpy, P is the pressure, J is the Jacobian of the transforma-
tion, M, is the free-stream Mach number, and y is the specific heat ratio. P,
p, and the velocity vector (u,v) are nondimensionalized by their values at the
free-stream condition, E and H are nondimensionalized by the free-stream inter-
nal energy pa&CvT, and other variables at t1' _ free-stream condition are corn-
puted from these variables. S
The physical boundary conditions are defined by the solid-wall condition
on the airfoil surface and the fact that the disturbances generated by the
airfoil propagate outward to infinity. The numerical implementation of these
physical boundary conditions will be discussed later. S
2.2 Numerical Scheme
A computational mesh is created by equally dividing the (X,Y) coordinates
in the mapped plane. In the mapped plane, the spatial derivatives in X at •
TN-215/09-84
-. - -" - -
-3- .
each mesh point are evaluated by application of a fast Fourier transform. The
derivatives in Y are evaluated by second-order central finite differences.
Evaluation of the elements in the transformation matrices (xXXy,yxyY is -
performed in the same manner. The singularity of the transformation at the
trailing edge is avoided by placing it between two mesh points.
By using this method of evaluating spatial derivatives, the governing
equations are converted to a system of ordinary differential equations in time.
These equations can be solved numerically by using any of a variety of well-
developed techniques for the solution of ordinary differential equations.
An approximate fourth-order Runge-Kutta scheme is used in this work. The
algorithm is given by the following equations:
-1(n) -1o 1 -(n-1)q +(5-n)- ; n 1 1, ., 4 (2)
4 .where q represents the flow variables at the mesh points, R represents the
terms with spatial derivatives in the equations, and n denotes the Runge-Kutta
step. For a linear wave equation, this scheme has been shown to be stable for
a CFL number less than 2.8 (Jameson, Schmidt, and Turkel, 1981) for a finite
difference scheme, and is stable for our hybrid scheme with a CFL number less
than or equal to 2. Following Jameson et al., a local time step that is re-
stricted by the CFL number is used. Because of this, no physical interpreta-
tion should be given to the transient solutions.
~.0.2.3 Filtering
Filtering is required in suppressing the Gibbs error. A Schumann filter
used by Gottlieb et al. (1982) and given by the following formula has been
applied every 35 time steps at a CFL number of 3.5 (see Section 2.5, Convergence •
Acceleration).
q o.25(+ 1 + + qK 1); K- ij (3)
where i and j denote indices of the discrete points in the X and Y directions,
respectively. At the shock wave, one-sided filtering in the X direction is
applied to preserve the sharpness of the shock wave. The Schumann filter is
equivalent to a first-order artificial viscosity. However, the filter is
applied only every 35 time steps. The order of the error is higher than first
order. No high-mode smoothing (Gottlieb et al., 1982) has been applied.
TN-215/09-84
6 W
-4-
We have experimented with other filtering methods, such as derivative
smoothing and artificial viscosity. The former is a low-pass filter in the
wave number space when evaluating derivatives spectrally. It is expected to
yield a smoother residue and thus stabilize the calculations. Numerical
experiments do not favor this approach, as the filter fails to stabilize the
computations. As to the artificial viscosity, the shock wave is smeared in
the computations, which defeats the nondissipative nature of the spectral
scheme.
2.4 Boundary Conditions •
The numerical implementation of boundary conditions for a hyperbolic
system of partial differential equations is an active research subject in
itself. Essentially, on a boundary Y0, there are four characteristics that
correspond to the speeds qn,2qt, qn - c and qn + c. The respective charac-
teristic variables are p - c0P, qt' p - 0oc0qn, and p + 0 c0qn, where the
subscript o stands for the quantities at the previous time step. Only the
characteristic variables carried on the outgoing characteristics from the
interior of the fluid domain can be computed from the governing equations. 9The characteristic variables carried on the incoming characteristics must be
replaced by the appropriate boundary conditions. The flow quantities can then
be recovered from the combination of the boundary conditions and the outgoing
characteristic variables.
On the solid surface Y = 0, there is only one incoming characteristic.
Let (M,N) be the momentum along the surface and normal to the surface,
respectively, and AQ be the symbol for the temporal change of physical
quantities Q as computed by the interior formula, with the subscript c
denoting the quantities computed by the interior formula. The following
formulae for the physical quantities at the boundary points can then be given.
P- AE + Y(Y-1) (- AM + . A) (4)
P P + AP (5)
( o/ + (AM - P'M/Po)/po (6)
TN-215/09-84
r
-5- p
S=p + AP (7)PC
-0p = Pc ; - M2 AN'co (8) SCI Cn) 2 (9).
(n) = (1 ) .'-'-..
M = p (n)(M/P) (10)
M) ( n )
E(n) = (n)+ y(Y-l) M 2 M /(n (n) (12) s
where the superscript n stands for the newly advanced quantities, and all
velocities are nondimensionalized by the free-stream velocity.
At the far field boundary, the treatment is essentially the same as that
used by Jameson et al. (1981) except that the "extrapolated" quantities as
defined in that paper are those computed by the interior computations.
2.5 Convergence Acceleration
To increase the stability of the time-stepping scheme, an additional
residue-smoothing" process (Jameson and Baker, 1983) has been implemented.
After the residue R has been evaluated at every mesh point, the residues are
smoothed by a linear transformation defined as follows: LO
= (1- 2-1 2 -10 S _(l - tS )_ R (13)X Y
where 6 and 6 are conventional finite difference operators in X and Y, and "x Yis the parameter for the residue-averaging process. The new modified residueCDfield R is used to advance the solution in time. This process alters the time-
dependent solution without changing its steady state. To bring out the essen-
tials of the residue-averaging process, a simple wave equation is considered:
*t + Cx = 0 . (14) p
The residue-averaging process as described is equivalent, to the lowest
order, to adding an additional term to the original simple wave equation and
converting it to the following equation:
x (Ax)2 txx 0 (15)
TN-215/09-84
C+ - _-
The dispersion relation for this equation can be given as
k 1+ek ) c(16)k k2(Ax) 2
where W is the frequency and k is the wave number. By increasing the para-
meter C, the wave speed for the high wave number component is substantially -
increased. This decrease in wave speed for the dangerous short waves contri- S
butes to the substantial increase in the time step. In fact, Equation (15) is
the linearized form of a model equation for long dispersive waves discussed by
Benjamin, Bona, and Mahony (1972), who pointed out the numerical advantage of
this equation over the Korteweg-de-Vries equation. Other means of manipulating S
the dispersion relation to gain stability have been suggested (e.g., Gottlieb
and Turkel, 1980). However, these methods do not recover the original equation
in steady state, although the error is of higher order. The residue-averaging
process substantially extends the stability boundary. A CFL number of 3.5 has
been used without any difficulty.
C
TN-215/09-84
* . ..
-7-
3. Computed Results for Hybrid Scheme
The initial effort employed a hybrid spectral/finite difference discretiza-
tion to gain some experience with the problem. In the circumferential direc-
tion, the variables are expanded in a Fourier series because of the periodic
nature of the problem. In the radial direction, a central finite difference "..-'
scheme is used. This hybrid scheme is designed to answer some questions in
applying the pseudospectral method to transonic flows. The application of the
pseudospectral method to a realistic airfoil shape can be demonstrated by this
scheme. Since we expect that the shock wave will be normal to the airfoil
surface, the discontinuity is mainly in the X direction. Adequate resolution •
of the shock wave can be achieved and the question of convergence of the
spectral series can be answered using the hybrid scheme. Other properties,
such as the time-stepping scheme, convergence acceleration, and the filtering
technique, can also be studied with the hybrid scheme. We shall use a Karman- •
Trefftz airfoil for this work because of its simple analytical mapping from
the physical plane to the interior of the circle. The method can easily be
extended to an airfoil of arbitrary shape by using a truncated complex series
to map the profile to a circle.
For testing the solution algorithm, flows around a circular cylinder are
computed. The pressure distribution for a subcritical flow with MC. 0.39
is given in Figure 1. The computation is performed on a 64x24 grid (64 points
in the circumferential direction, 24 points radially). It has been computed
without filtering. A supercritical case with M. 0.45 is also computed,
and the results are shown in Figure 2. Filtering is performed every 35 time
steps with a CFL number of 3.5. The results agree with a finite volume cal-
culation by Jameson et al. (1981). The shock wave has no internal structure
and is sharply defined.
A Karman-Trefftz airfoil with the following transformation from the mapped
plane C to the physical plane z is chosen for calculations.
z__ (1+Lw + (1-L)1
(1+L)K) + (I-L )
2 1/2L (I no) -t ;C (ono 0 (-0.1,0) (18)
TN-215/09-84
--
Supercritical non-lifting flows with MW - 0.75 are computed on a
* 64x24 grid. The results are shown in Figure 3, together with the results from
a finite volume calculation. The hybrid calculation shows again a sharply
defined shock wave. The agreement between the two calculations is very good.
In particular, the positions of the shock as defined by the midpoint of the
* structure show close agreement. There are discrepancies immediately behind
the shock wave, however, and the source of these discrepancies is not clear.
The pressure ratio across the shock wave using the pseudospectral calculation
has been checked against that using the Rankine-Hugoniot relation based on the
upstream Mach number. The error is less than 4 percent. 0
To demonstrate the convergence of the Fourier series, the same case is
computed on a 32x24 grid. The results are shown in Figure 4, together with
the results of calculations on a denser mesh using a finite volume calcula-
C tion. The accuracy of the 32x24 calculation is quite good. The shock
resolution of the sparse mesh calculation is comparable to that of the 64x24
finite volume calculation. As expected, the finite volume calculation on the
sparse grid does not produce acceptable results.
TN-215/09-84
-9-
4. Full Spectral Scheme
To carry the work further, we have attempted to construct a full spectral .
method. Instead of using a finite difference method for the radial direction
in the transformed circular plane, we have used a Chebyshev polynomial -..-'-"-* -
expansion for that direction.
We immediately encountered difficulties, however, in using the Chebyshev
spectral method. The first difficulty is the stability problem. Chebyshev
collocation points are defined as
y [1 cos J (19)
where N+l > J > 1 is the computational coordinate and 1 > y • 0 is the physiLcal
space. It is easy to verify that the spacing of the grid points at the end
points is asymptotically
Ay 0 (• (20)
For viscous problems, the end points are usually on the solid surface, where
the flow -,elocity vanishes. The time steps can maintain a reasonable size
even though the grid spacing is small, since
At - CFL . (21)
In the present case,
At CFL (22)
where c is the speed of sound. The small grid size near the end points forces
At N2 .(23)
N2 -
The solution will take an excessively large number of time steps to develop.
Also, because of large variations in the grid spacing, the local time step
approach used successfully in the hybrid method does not seem to apply.
An attempt to stretch the Chebyshev grid to achieve a more uniform grid
spacing also failed. The accuracy near the end points deteriorates, which
affects the accurate application of boundary conditions. This deterioration
of accuracy in a stretched grid can be understood by taking the extreme
example of restoration to a uniform grid. Let
r - (1- cos wy) (24)
TN-215/09-84
-10-
where the computational space n is divided into Chebyshev collocation points
and the corresponding y coordinate is uniform. By the chain rule, we have
• Sa dy a (25)3y --dy 3i -
ILI sin Wy (26)dy 2
Since dn/dy approaches zero at the end points, accurate evaluation of the
derivative at the end points is not possible.
We currently plan to investigate two other methods. The first is to
discard the Chebyshev method and attempt the Fourier polynomial subtraction
method (Gottlieb and Orszag, 1977). The second is to develop an implicit
method to circumvent the stability problem.
• p.
C"
T2/ 8( TN25/98
5. Conclusions
The hybrid method shows promise for the use of spectral methods to accur-
ately resolve the shock wave in a realistic problem. However, the filtering
of the solution certainly deteriorates the accuracy. Furthermore, for complex
three-dimensional problems, it is difficult to identify a supersonic-supersonic
shock. Present means of identifying shock waves by the sonic condition do not
apply there. Unlike the finite difference method, the residue of a spectral
method does not decrease with the number of time steps. At present, the
number of supersonic points is used as the indicator of convergence.
The full spectral method using a Chebyshev expansion also presents diffi-
culties. Further investigations of alternative methods are required to assess
the merit of a full spectral scheme.
C..
( S°
2612R
TN-215/09-84 .
-12- S
References
Benjamin, T. B., Bona, J. L., and Mahony, J. J. (1972) "Model Equations forLong Waves in Non-Linear Dispersive Systems," Phil. Trans. Roy. Soc. S
London (Ser. A.), 272, pp. 47-78.
Gottlieb, D., Lustman, L., and Streett, C. L. (1982) "Spectral Methods forTwo-Dimensional Shocks," ICASE rep. 82-38.
Gottlieb, D., Lustman, L., and Orszag, S. A. (1981) "Spectral Calculations of @One-Dimensional Inviscid Compressible Flows," SIAM J. Sci. Stat.Comput., 2, 3, pp. 296-310.
Gottlieb, D., and Turkel, E. (1980) "On Time Discretization for SpectralMethods," Stud. Appl. Math., 63, pp. 67-86.
Gottlieb, D., and Orszag, S. (1977) "Numerical Analysis of Spectral Methods:
Theory and Applications," NSF-CBMS Monograph No. 26, Society of Industrialand Applied Mathematics.
Jameson, A., and Baker, T. (1983) "Solution of the Euler Equations for Complex
Configurations," AIAA paper 83-1929. .0
Jameson, A., Schmidt, W., and Turkel, E. (1981) "Numerical Solutions of theEuler Equations by Finite Volume Methods Using Runge-Kutta Time-SteppingSchemes," AIAA paper 81-1259.
Jou, W. H., Jameson, A., and Metcalfe, R. W. (1983) "Pseudospectral Calcula- •tions of Two-Dimensional Transonic Flow," Proceedings of Fifth GAMMConference on Numerical Methods in Fluid Mechanics, Rome, Italy,October 5-7.
Voight, R. G., Gottlieb. D., and Hussaini, M. Y. (1984) Spectral Methods for
Partial Differential Equations, Society for Industrial and AppliedMathematics, Philadelphia, 267 pp.
TN-215/09-84
-1. .
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, -; ; : + . :......*.-.-....-....,.-.', + -.- ..... +.-.+.,....- -.,-. -.... ,..,-._ ...,
C. . ,. . . ... _. _ ,_,._ . ,. _ ,,, . ..r:,,,I,.L.. . , +. . . - , . -. .• -. . . . - . - . ,- . '. ,. , . . - - ,,,.. .+,..
79
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Appendix
C "Pseudospectral Calculations of
Two-Dimensional Transonic Flow"
* (Presented at Fifth
GAMM Conference on Numerical Methods
in Fluid Mechanics, Rome, Italy,
October 5-7, 1983.)
( TN-215/09-84
PSEUDOSPECTRAL CALCULATIONS OF TWO-DIMENSIONALTRANSONIC FLOW
Wen-Huei JouFlow Research Company, Kent, Washington, U.S.A. 0
Antony JamesonPrinceton University, Princeton, New Jersey, U.S.A.
Ralph MetcalfeFlow Research Company, Kent, Washington, U.S.A.
SUMMARY
A hybrid pseudospectral-finite difference scheme is used tocalculate transonic flow over a two-dimensional object using theEuler equations. The exterior of the object is mapped to theinterior of a circle. The flow field variables are discre- 3tized using a Fourier series in the circumferential direction,while a central finite difference scheme is used in the radialdirection. We used a four-stage Runge-Kutta scheme including afilter and a residue-smoothing process. Transonic flows over acircular cylinder as well as a Karman-Trefftz airfoil werecomputed. The results are compared to those from finite volume 5calculations. It is found that the pseudospectral calculationsare able to produce shocks with no internal structure, and fewergrid points are needed to obtain the required, accuracy.
INTRODUCTION
In recent years, there has been strong interest in computa-tions of transonic flows using the time-dependent Eulerequations. This interest stems in part from the possibility ofshock-generated vorticity in the flow field and in part from theinterest in numerical methods for a nonlinear hyperbolicsystem. Most of the numerical methods are finite difference innature and are second order in accuracy. To stabilize thecomputation and to smooth the dispersive error for unsteadycomputations, either artificial dissipative terms are added to
the equations or a built-in dissipative mechanism is included inthe numerical scheme. These dissipative terms usually cause theshock wave to span across three to four grid points. To capturea shock with reasonable accuracy, one is forced to use fairlydense grid distributions over the region where the shock wave isexpected to be.
The pseudospectral method is an alternative to the finitedifference method. It has been applied successfully to manysmoothly varying flows. The numerical analysis of the methodhas been given in detail by Gottlieb and Orszag in a monograph[1]. Because of its high rate of convergence, the methodusually requires relatively few terms of the basis functions foraccurate computations. In addition to the spatial accuracy, thedispersive error for unsteady computations is also minimized.
Recently, efforts have been made to apply the pseudo-spectral method to flows with shock waves. Gottlieb, Lustman,and Orszag [2] have demonstrated the feasibility of the pseudo-spectral method through the solution of a one-dimensional shock
S114/09-83
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tube problem. By using the shock-capturing technique, theyshowed that the shock wave can be resolved within one grid point.The Gibbs phenomenon error due to the discontinuity can be fil-tered to improve the accuracy. Gottlieb, Lustman, and Streett 0[3) have attempted the two-dimensional problem of the reflectionof an oblique shock from a wall. The results from this investi-gation are encouraging. Using a fairly sparse grid, they showedthat the shock wave can be resolved within one grid point. Theaccuracy of the solution as compared to the exact solution isreasonable considering the sparseness of the grid points.
In the present work, we shall consider steady transonic flowsaround an airfoil by solving the Euler equations. We shall ad-dress problems of applying the pseudospectral method to flowsaround a complex geometry, including the development of a time-stepping scheme, and enhancement of the stability by residueaveraging. Numerical experimentation has been used to confirm 9convergence with a small number of basis functions, and also thecapability to treat shock waves with the aid of filtering.
GOVERNING EQUATIONS AND BASIC APPROACHr.0
The basic approach is to map the exterior of an airfoil tothe interior of a circle. Polar coordinates in the mapped planewill be used. Spectral decomposition of the solution can beused in the mapped plane. To serve this purpose, the Eulerequations in the mapped plane are written in fully conservativeform by using both physical and contravariant velocities. These _equations are
(q) + + ~ 0()ax aywhere E pU 2 PV '/y
2
SPUU +yyp/-YMOI PUV -.[- .
q1 v] OVU - xYP/Y M + x P/YM X0
[~ =[- Y] 1 H 4(y 1 lP + EJ /P
(x,y) are Cartesian coordinates in the physical plane, (XY) arepolar coordinates in the mapped plane, p is the density, (u,v)are velocity components, (U,V) are unscaled contravariantvelocity components, E is the energy, H is the specificenthalpy, P is the pressure, J is the Jacobian of the trans-formation, Mw is the free-stream Mach number, and y is thespecific heat ratio. P, p and the velocity vector (u,v) arenondimensionalized by their values at the free-stream condition, OE and H are nondimensionalized by the free-stream internal energypwCvT=, and other variables at the free-stream condition arecomputed from these variables.
The physical boundary conditions are defined by the solid-wall condition on the airfoil surface and the fact that the
114/09-83
disturbances generated by the airfoil propagate outward toinfinity. The numerical implementation of these physicalboundary conditions will be discussed later.
This initial effort employed a hybrid spectral-finite*difference discretization to gain some experience with the4
problem. In the circumferential direction, the variables areexpanded in a Fourier series because of the periodic nature ofthe problem. In the radial direction, a central finitedifference scheme is used. This hybrid scheme is designed toanswer some questions in applying the pseudospectral method to
* transonic flows. The application of the pseudospectral methodto a realistic airfoil shape can be demonstrated by this scheme.Since we expect that the shock wave will be normal to the air-foil surface, the discontinuity is mainly in the X direction.Adequate resolution of the shock wave can be achieved and thequestion of convergence of the spectral series can be answered
* using the hybrid scheme. Other properties, such as the time-stepping scheme, convergence acceleration, and the filteringtechnique can also be studied with the hybrid scheme. We shalluse a Karman-Trefftz airfoil for this work because of its simple
the circle. The method can easily be extended to an airfoil of
(arbitrary shape by using a truncated complex series to map theI'profile to a circle.
NUMERICAL SCHEME
* A computational mesh is created by equally dividing theMXY) coordinates in the mapped plane. In the mapped plane,the spatial derivatives in X at each mesh point are evaluatedby application of a fast Fourier transform. The derivatives inY are evaluated by second-order central finite differences.Evaluation of the elements in the transformation matrices
*(xX,xyyX,yy) is performed in the same manner. The singularity '7of the transformation at the trailing edge is avoided by placingit between two mesh points.
By using this method of evaluating spatial derivatives, thegoverning equations are converted to a system of ordinary dif-ferential equations in time. These equations can be solvednumerically by using any cf a variety of well-developed techni-ques for the solution of ordinary differential equations. Anapproximate fourth-order Runge-Kutta scheme is used in this -
work. The algorithm is given by the following equations:
*(n) -0 1 V(n-l) n 1, .. 4(2q q +(T5 -n7 n e 2
where q represents the flow variables at the mesh points, R Xirepresents the terms with spatial derivatives in the equations,and n denotes the Runge-Kutta step. This scheme has been shownto be stable for a CFL number less than 2.8 [4] for a finitedifference scheme, and is stable for our hybrid scheme with aCFL number less than or equal to 2. Following Jameson, Schmidt,and Turkel [4), a local time step that is restricted by the CFLnumber is used. Because of this, no physical interpretation *~
should be given to the transient solutions.
114/09-8 3
FILTERING
Filtering is required in suppressing the Gibbs error. ASchumann filter used by Gottlieb, Lustman, and Streett [3) and
given by the following formula has been applied every 35 timesteps at a CFL number of 3.5 (see later section on convergenceacceleration).
q= 0.25(qKl + 2q + qK-)r K = i,j (3)
where i and j denote indices of the discrete points in the Xand Y directions, respectively. At the shock wave, one-sidedfiltering in the X direction is applied to preserve the sharp-ness of the shock wave. The Schumann filter is equivalent to afirst-order artificial viscosity. However, the filter isapplied only every 35 time steps. The order of the error ishigher than first order. No high-mode smoothing [3] has beenapplied.
BOUNDARY CONDITIONS .
The numerical implementation of boundary conditions for ahyperbolic system of partial differential equations is anactive research subject in itself. Essentially, on a boundaryYo, there are four characteristics that correspond to thespeeds qn, qt, qn - c and qn + c. The respective characteristicvariables are p - cap, qt, p - pocoqn, and p + oocoqn , wherethe subscript o stands for the quantities at the previous timestep. Only the characteristic variables carried on the outgoingcharacteristics from the interior of the fluid domain can becomputed from the governing equations. The characteristicvariables carried on the incoming characteristics must bereplaced by the appropriate boundary conditions. The flowquantities can then be recovered from the combination of theboundary conditions and the outgoing characteristic variables.
On the solid surface Y = 0, there is only one incomingcharacteristic. Let (M,N) be the momentum along the surfaceand normal to the surface, respectively, and AQ be the symbolfor the temporal change of physical quantities 0 as computed bythe interior formula, with the subscript c denoting thequantities computed by the interior formula. The following . -
formulae for the physical quantities at the boundary points can .
then be given.AP= AE + y(y-l)M ( ! AM L L AP (4)
PC = P + AP (5)0
- Mo/Po + (AM - M /P U (6)
PC o + AP 7
P(n) -C YM. N.c (8)
114/09-83
U'~~. . . . . . . ..U - U
D(n) c + 2"-- (Pn _ pc)/C 20 (9) :-..YM2
(ni) (n).0M = n (M/P)c (10)
N = 0 (11)
(n P n + 2 ~-)M [M(n)J/p(n) (2E 2 (Y-l) OD (12) e
where the superscript n stands for the newly advanced quantities,and all velocities are nondimensionalized by the free-streamvelocity.
At the far field boundary, the treatment is essentially the - -
same as that used by Jameson, Schmidt, and Turkel [4] exceptthat the "extrapolated" quantities as defined in that paper arethose computed by the interior computations.
CONVERGENCE ACCELERATION
To increase the stability of the time-stepping scheme, anadditional "residue-smoothing" process [5) has been implemented.After the residue R has been evaluated at every mesh point, theresidues are smoothed by a linear transformation defined asfollows:
2 -1 2 -1( - 6 x) (1 - C6y) R (13)
where 6X and Sy are conventional finite difference operatorsin X and Y, and c is the parameter for the residue-averagingprocess. The new modified residue field R is used to advancethe solution in time. This process alters the time-dependentsolution without changing its steady state. To bring out theessentials of the residue-averaging process, a simple waveequation is considered:
t + co = 0 . (14)
The residue-averaging process as described is equivalent.to the lowest order, to adding an additional term to theoriginal simple wave equation and converting it to thefollowing equation:
t + cx - (x) txx 0 (15)
The dispersion relation for this equation can be given as
= 2 (16)1 + ek (6x)
where w is the frequency and k is the wave number. By increasing 0-the parameter c, the wave speed for the high wave number com-ponent is substantially increased. This decrease in wave speedfor the dangerous short waves contributes to the substantialincrease in the time step. In fact, Equation (15) is thelinearized form of a model equation for long dispersive waves
114/09-83
V.--..-- -]
discussed by Benjamin, Bona, and Mahony [6], who pointed outthe numerical advantage of this equation over the Korteweg-de-Vries equation. Other means of manipulating the dispersionrelation to gain stability have been suggested (e.g., [7)).However, these methods do not recover the original equation insteady state, although the error is of higher order. Theresidue-averaging process substantially extends the stabilityboundary. A CFL number of 3.5 has been used without anydifficulty.
COMPUTED RESULTS
For testing the solution algorithm, flows around a circularcylinder are computed. The pressure distribution for a sub-critical flow with M. = 0.39 is given in Figure 1. The com-putation is performed on a 64x24 grid (64 points in the Scircumferential direction, 24 points radially). It has beencomputed without filtering. A supercritical case withMw = 0.45 is also computed, and the results are shown inFigure 2. Filtering is performed every 35 time steps with aCFL number of 3.5. The results agree with a finite volumecalculation by Jameson, Schmidt, and Turkel, [4]. The shockwave has no internal structure and is sharply defined.
A Karman-Trefftz airfoil with the following transformationfrom the mapped plane C to the physical plane z is chosen forcalculations.
z_ = (I+L) + (-L) K = 1.9 (17) -
,L (l+L4VK + (l-LO)K
2 1/2 -.-.-L = (1 -n ) 1/2 - o = (Eo #o) = (-0.1,0) (18)
Supercritical non-lifting flows with M, = 0.75 are computedon a 64x24 grid. The results are shown in Figure 3, togetherwith the results from a finite volume calculation. The hybridcalculation shows again a sharply defined shock wave. Theagreement between the two calculations is very good. Inparticular, the positions of the shock as defined by themidpoint of the structure show close agreement. There arediscrepancies immediately behind the shock wave, however. Thesource of these discrepancies is not clear. The pressure ratioacross the shock wave using the pseudospectral calculation hasbeen checked against that using the Rankine-Hugoniot relationbased on the upstream Mach number. The error is less than4 percent.
To demonstrate the convergence of the Fourier series, thesame case is computed on a 32x24 grid. The results are shownin Figure 4, together with the results of calculations on a ......
denser mesh using a finite volume calculation. The accuracy ofthe 32x24 calculation is quite good. The shock resolution ofthe sparse mesh calculation is comparable to that of the 64x24 .finite volume calculation. As expected, the finite volumecalculation on the sparse grid does not produce acceptableresults.
114/09-83* - a * .~**. *** * * * . * t.- . -
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Fiur 1. Prsuedsr-Fgr . Pesr iti
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Figure 1. Pressure distri- Figure 2. Copresse dosribution for a subrinticalt beut forh a fiprciticflroive a Mclhe at7 fsngvlowe ovher afylnr atMac 0.394 Mach 0.45es)Tezaifi t ah07
*I114/09-8
CONCLUSIONS AND ACKNOWLEDGEMENTS
Several conclusions can be drawn from the presentinvestigation.
(1) In computing flows with shock waves, the Gibbs error can befiltered to produce accurate results. Because of the rapidconvergence of the Fourier series, fewer grid points arerequired than with the lower order difference-type scheme.
(2) A shock wave without internal structure can be produced. SThis capability also contributes to the accuracy of themethod in that fewer grid points are required to resolvethe shock wave.
(3) Application of the pseudospectral method to flows around arealistic geometry is possible using a mapping technique.
The authors are indebted to Mr. Morton Cooper of FlowResearch Company for suggesting the problem. Discussions withProfessor Steven Orszag have also been very helpful. This workwas supported by the Air Force Office of Scientific Researchand the Office of Naval Research under Contract No.F49620-82-C-0022.
REFERENCES
[1] Gottlieb, D., and Orszag, S., Numerical Analysis of SpectralMethods: Theory and Applications, NSF-CBMS Monograph No. 26,Society of Industrial and Applied Mathematics (1977).
[2) Gottlieb, D., Lustman, L., and Orszag, S. A., "Spectral cal-culations of one-dimensional inviscid compressible flows,"SIAM J. Sci. Stat. Comput., 2, 3 (1981) pp. 296-310. - -
[3) Gottlieb, D., Lustman, L., and Streett, C. L., "Spectralmethods for two-dimensional shocks," ICASE rep. 82-38(1982).
[4) Jameson, A., Schmidt, W., and Turkel, E., "Numerical solu-tions of the Euler equations by finite volume methods usingRunge-Kutta time-stepping schemes," AIAA paper 81-1259(1981).
[5) Jameson, A., and Baker, T., "Solution of the Euler equationsfor complex comfigurations," AIAA paper 83-1929 (1983).
[6) Benjamin, T. B., Bona, J. L., and Mahony, J. J., "Modelequations for long waves in non-linear dispersive systems,"Phil. Trans. Roy. Soc. London Ser. A., 272 (1972) pp. 47-78. ..-
[7) Gottlieb, D., and Turkel, E., "On time discretization forspectral methods," Stud. Appl. Math., 63 (1980) pp. 67-86.
114/09-83
Flow Technical Note No. 216
NUMERICAL INVESTIGATION OF
VTOL AERODYNAMICS*
0
by
M. H. Rizk
September 1984
Flow Industries, Inc.Research and TechnologyKent, Washington 98032
*This work is supported by the Air Force office of Scientific Research underContract No. F49620-82-C-0022.
Table of Contents
Page
List of Figures i
1. Introduction 1
2. Governing Equations 3
3. Method of Solution 5
4. Numerical Examples 6
Figures 9
TN-21/09-8
17 i
-ii-
List of Figures
Page
Figure 1. Model Problem for a Hovering VTOL Aircraft 9
Figure 2. Schematic of Flow Field about a Hovering VTOL Aircraft 10
Figure 3. Example 1: Velocity Vectors in the Plane x x. 11Fu 4
Figure 4. Example 1: Velocity Vectors in the Plane x xf 12
Figure 5. Example 1: Velocity Vectors in the Plane y = yd 12Figure 6. Example 1: Velocity Vectors in the Plane y = Yd 13..
Figure 7. Example 1: Pressure Contours in the Plane x x. 14
F r 8Figure 8. Example 1: Pressure Contours in the Plane y y 15
Figure 9. Example I: Pressure Contours in the Plane z = Z 16
Figure 10. Example 1: Pressure Contours in the Plane z za 16
Figure 11. Example 1: x-Vorticity Component in e Plane x = x. 17JFigure 12. Example 1: y-Vorticity Component in the Plane y = yj 18
Figure 13. Example 2: Velocity Vectors in the Plane x = x. 19
Figure 14. Example 2: Velocity Vectors in the Plane x -*x 19f
Figure 15. Example 2: Velocity Vectors in the Plane y yj 20
Figure 16. Example 2: Velocity Vectors in the Plane y Y Yd 21
Figure 17. Example 2: Pressure Contours in the Plane x x. 22Figure 18. Example 2: Pressure Contours in the Plane y yj 23.
Figure 19. Example 2: Pressure Contours in the Plane z z z 24
Figure 20. Example 2: Pressure Contours in the Plane z = z 24gFigure 21. Example 2: x-Vorticity Component in the Plane x = x. 25
Figure 22. Example 2: y-Vorticity Component in the Plane y = yj 26
Figure 23. Example 3: Velocity Vectors in the Plane x - x . 27F r 2
Figure 23. Example 3: Velocity Vectors in the Plane x = x. 27Figure 24. Example 3: Velocity Vectors in the Plane y = f 27
Figure 25. Example 3: Velocity Vectors in the Plane y = Yd 28.:":....
Figure 27. Example 3: Pressure Contours in the Plane x x. 30
Figure 28. Example 3: Pressure Contours in the Plane y yj 31Figure 29. Example 3: Pressure Contours in the Plane z - z 32
Figure 30. Example 3: Pressure Contours in the Plane z a 32a
Figure 31. Example 3: x-Vorticity Component in the Plane x = x. 33
Figure 32. Example 3: y-Vorticity Component in the Plane y y 34
4 t TN-216/09-84 a
............................................
- . - . . f - r
-iii-
List of Figures
Page
Figure 33. Example 4: Velocity Vectors in the Plane x = x. 35
Figure 34. Example 4: Velocity Vectors in the Plane x = xf 35
Figure 35. Example 4: Velocity Vectors in the Plane y = y. 36J0
Figure 36. Example 4: Velocity Vectors in the Plane y = d 37
Figure 37. Example 4: Pressure Contours in the Plane x = x. 38
Figure 38. Example 4: Pressure Contours in the Plane y - yj 39
Figure 39. Example 4: Pressure Contours in the Plane z = z 40
Figure 40. Example 4: Pressure Contours in the Plane z = z 40a""
Figure 41. Example 4: x-Vorticity Component in the Plane x = x. 41
Figure 42. Example 4: y-Vorticity Component in the Plane y = yj 42F u ..ti-
Figure 43. Example 5: Velocity Vectors in the Plane x = x. 43u 4
Figure 44. Example 5: Velocity Vectors in the Plane x = xf 43
Figure 45. Example 5: Velocity Vectors in the Plane y = yx 44Figure 46. Example 5: Velocity Vectors in the Plane y = d 4
Figure 47. Example 5: Pressure Contours in the Plane x = x. 46
Figure 48. Example 5: Pressure Contours in the Plane y y. 47io
Figure 50. Example 5: Pressure Contours in the Plane z = za 49
Figure 51. Example 5: x-Vorticity Component in the Plane x = x. 50
Figure 52. Example 5: y-Vorticity Component in the Plane y = yj 51Figure 53. Initial Vortex Propagation in Example 5 52'Figure 54. Initial Vortex Propagation in Example 5 53
C 0%
TN-216/09-84
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m -I 'jm.. . . . . . . . . . ...D ' ° "#,a • o o" *. ". ". •
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-1-
1. Introduction
Work is in progress at Flow Industries on the direct numerical simulation
of complex VTOL flows using the full three-dimensional, time-dependent
Navier-Stokes equations. The objective of this numerical simulation is to
compute accurately the details of the flow field and to achieve a better
understanding of the physics of the flow, including the role of initial
turbulence in the jet, the influence of forward motion on hover aerodynamics,
the collision zone and fountain characteristics, and the jet structure and
entrainment process in the transitional flight regime. The results of this
work can be used to evaluate the merit of various models suggested in the past -
or can be used to construct a new model. This work will also allow the
assessment of wing-jet-ground interference effects and the accurate prediction
of their associated forces and moments, which is required for the design and
optimization of VTOL aircraft. This note describes the work completed at Flow
in the second year of a program in which VTOL aerodynamics are being
investigated numerically.
The problem under investigation is that of an infinite row of jets
impinging on the ground (see Figure 1). This problem, which contains the
essential features of twin jets impinging on the ground (see Figure 2),
simulates the hovering configuration. The choice of a row of jets provides .-.
the periodic property of the flow field, which allows approximation of the .
flow properties in the periodic direction by a truncated Fourier series. The
spectral method may therefore be used in the periodic direction, while finite
difference approximations are used in the vertical z direction and the
y direction normal to the row of jets. The jets may be inclined in the
y direction, which leads to a configuration associated with an aircraft in
pitch while hovering. By imposing a cross flow in the y direction, it is
possible to study the effects of the aircraft's forward motion during takeoff
and transition.
A computer code that solves the time-dependent Navier-Stokes equations has
been developed with the purpose of numerically simulating the problem of an
infinite row of jets impinging on the ground. The code presently uses finite
difference approximations in all three spatial directions, and it uses a
first-order time-differencing scheme. Modifications are in progress that will
allow the use of the spectral method in the periodic direction and the use of
TN-216/09-84 0
.§:9~:: 2:§*~:~X9~,:--.:-.:../.
-2-
a second-order time-differencing scheme. Subgrid-scale modeling, which allows
the solution of problems at high Reynolds numbers, will also be introduced
into the code. Although the code is not in its final form, it has been used
to obtain solutions that indicate the main features of VTOL aerodynamics. "-
In this note, the governing equations and the boundary conditions used in
the code are summarized in Section 2. The method of solution is discussed in-
Section 3, and preliminary examples of solutions using the code are presented
in Section 4.
.O
S~ . . . .-. - ,
-3-
2. Governing Equations
The governing equations are the Navier-Stokes equation -
ReV V2 (1)i~
2t+ (qeV)q -- Vp 72q
and the continuity equation _.. 0 (2)V "q - 0 (2) 0
where q is the velocity vector and p is the pressure. When the Reynolds
number is too large to resolve numerically the entire range of energetic scales,
filtering will be used to eliminate the smaller (subgrid-scale) motions. •
Filtering Equation (1) introduces new terms, similar to Reynolds stress terms
obtained in the Reynolds-averaged equations, that contain the effect of the
subgrid-scale motions on the numerically resolved motions. We plan to use
standard procedures to handle these terms and will introduce them into our -
numerical scheme at a later date. By taking the divergence of Equation (I),
the following Poisson equation governing the pressure is-obtained:
V2p -V'[(q'V)q] - (V'q) + _ (V.q) (3)- .- t Re 9 3
Substituting Equation (2) into Equation (3) leads to the following pressure .. ,
equation:
V2p l -V[(qV)q] . (4) S
The system of Equations (1) and (4) is equivalent to the original system,
Equations (1) and (2), and is used here instead of the original set of
equations. The vector equation (1) is solved subject to the periodicity
condition in the x direction; a weak outflow condition
au ay - o , ~(5).. :-
Aith av/ay being determined from the continuity equation, is applied at
the side boundaries of the computational domain (y YB' y = Yb) ' and a
no-slip condition is applied at the bottom boundary, z = z g' and the top .
90
T-216/09-84
4• -2k e .I* - -
• . • . -
-4-
boundary, z -z a' outside the jet region. In the jet region, the inflow
condition
q(x,y,za ) = f(x,y) (6)
is specified. Equation (4) is solved subject to the condition
0 (7)
at the side boundaries and subject to the Neumann boundary conditions
determined from Equation (1) at the top and bottom boundaries.
T /
0•;.il
................................. ..........................
-o -" -.- --" -." -. " --" ".," --". ...- "-'- tt.... .& '" " -- -"- - " -" " - - - -"-"- a- '2 C" -" " -"" '. -" - % "-' -" ' ."-- - .- " -i- ' -
-5-
3. Method of Solution
The finite difference approximations to Equations (1) and (4) are written .
at the mesh points of a staggered grid. For advancing the solution from timeto time tn n+lt to time t , where tn - nAt, t = (n+l) At and At is the time step, the - - -
following first-order scheme is used:
n+l qn + q n - + V q2 n (8) 0q =q+At[-(qnV)q - qp+-- .. . -Re
n n+lwhere p is determined so that mass conservation is assured at time t
It is determined by solving the finite difference approximation to the equation .0
V~p = -V-[(qn'vlqn] (9)
This equation is solved by using a direct (noniterative) fast Laplace equation
solver. 0
T •_. /09°8.
TN-2160.-84- o
*...-... . . . . . ... % % ~ ' N .' * - . . . .. . . . . . . . . ...
-6-
4. Numerical Examples
The examples presented here are preliminary examples that have been solved B
using the developed computer code. A relatively coarse numerical mesh is used,
and the Reynolds number is assumed to be low enough so that filtering is not
required. The results presented here are not intended to be an accurate
simulation of VTOL flow configurations. Nevertheless, they do indicate the
main features of these flows.
For all the examples presented here, the plane y = yj is assumed to be a
plane of symmetry. Unless otherwise stated, the computational domain is
defined by (see Figure 1)
0 = x < x x = 1
-2 = B y 1 Yb = 2
O=zg <z a =<
where all dimensions are normalized by the jet diameter. The jet velocity
profile in the direction of the jet axis is assumed to be given by
2Qj(r) = 1I rR 2
where R. is the jet radius, r is the distance from the jet axis and velocities
are normalized by the jet maximum velocity. The Reynolds number in the
examples is based on the jet diameter and the maximum jet velocity.
Example 1:
In this example the jet axis is assumed to be normal to the ground plane
(a = 90*) and there is assumed to be no cross flow (V = 0). The Reynoldsnumber is given by Re = 300. An 18x72x18 (x,y,z) mesh is used.
Figures 3 through 12 show the main features of the flow generated by a row
of vertical jets impinging on the ground. The velocity vectors in the planes
x = xj, x = x are shown in Figures 3 and 4, respectively. The fan-shapedf
fountain that results from the collision of the two wall jets is apparent in S
Figure 4. The jet, the wall jet and the fountain are apparent in Figure 5.
In both Figures 5 and 6 a downward motion exists as the plane x = x. is
approached, while an upward motion exists as the plane x = x is approached.
However, the relative magnitudes of these motions are reversed in the two
TN-216/09-84 S
-7-
figures. Figure 5 indicates that the downward motion in the jet is greater
than the upward motion in the fountain. The plane of Figure 6 does not pass -S
through the jet. There, the fountain upward motion is relatively stronger
than the downward motion at x = x. Figures 7 through 10 are pressure.,
contours that indicate high-pressure areas in the zones of jet-ground
impingement, wall jet-wall jet collision and fountain impingement on the upper
boundary. Figures 11 and 12 are contour plots for the vorticity components.
Example 2:
In this example the jet axis is assumed to be inclined at an angle
a = 600 to the ground. A cross flow of V = 0.2 is also assumed. The
Reynolds number is given by Re = 300. An 18x72x18 (x,y,z) mesh is used.
Figures 13 through 22 show the main features of the flow generated by a
row of inclined jets impinging on the ground in a cross flow. In Figure 13
the ground vortex formed by the interaction of the cross flow and the wall jet
is apparent. The effect of the cross flow on the fan-shaped fountain is shown
in Figure 14, where it is no longer symmetric.
*|
For the problem of a jet in a cross flow, two basic configurations are
relevant to VTOL aerodynamics. In the first configuration, the jet impinges
on the ground. The main features of this flow are indicated in Example 2. A
second configuration results as the distance between the aircraft and the
ground becomes large and/or as the forward aircraft speed becomes large. In
this case, the jet does not impinge on the ground. This configuration is used
in the following example.
Example 3:
In this example a - 900, V = 0.7 and Re = 60. A 7x28x14 mesh is used.
The computational domain is defined by .]0 =x. < x < Xf 1'
-2 = YB y < Yb = 2
0 nz < z <z -20 g- - a I .,.
Figures 23 through 32 show the main features of this flow. Figure 23
indicates that the jet changes its direction before it reaches the ground. As .-40
TN-216/09-84
................................................,% " ° % % ,% , , ,°o,- , '° % % ,, •% . ,, , % . o' " • °%%"""""'".".. . . . . ..-. . . ...= * "C ., o. . ° . .'' -o . ,. .
-8-
indicated in Figure 24, no fountain flow develops in this example since there
are no wall jets. The double vortex generated by the jet-cross flow interac-
tion is shown in Figure 26. As indicated by the pressure contours shown in .
Figure 27, a high-pressure region develops upstream of the jet, while a
low-pressure region develops downstream of the jet in its wake.
Example 4:
This example is the same as Example 3 with the exception that V 0.
Figures 33 through 42, which show the main features of the flow for this case,
thus allow comparison between the zero cross-flow case (given here) and a
cross-flow case (Example 3) for the same configuration.
Example 5:
This example is similar to Example I (a = 900, V 0). However, a
relatively fine computational mesh is used here, which allows the use of a
relatively large Reynolds number (Re = 600). Symmetry is assumed in the x
direction in addition to the y direction. Therefore, the computational domain
is defined by
0 x. < X < xf = 1
0 ff yj y yb 2
0 z <z<z = 1g - -- a ....
A 24x48x24 computational mesh is used.
The results of this calculation are indicated in Figures 43 through 52.
Qualitatively similar effects to those observed in the first example are
indicated. However, certain effects, such as the propagation of the initial
vortex, while observed for the high-Reynolds-number, fine-mesh calculation
(see Figure 53) are not observed for the low-Reynolds-number, coarse-mesh
calculation (see Figure 54).
2614R
TN-216/09-84
-9-
ZS
SIDE VIEW
* F:
END VIEW
Figure 1. Model Problem forma Hovering VTOL Aircraft
100
LL
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Figure 3. Example 1: Velocity Vectors in the Plane x = xj
, Y..
. . . . . . . . , ' % i I I / e t .
I I IYB Yj Yd Yb
Figure 4. Example 1: Velocity Vectors In the Plane x =xf
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Figure 5. Example 1: Velocity Vectors in the Plane y = yj ...
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(-17-
z
y* ______________ ____________
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I
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z0
-18-
zaz
I f
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Figue 1 . Ex mpl 1:y-Voticty ompo entin he Paney y
-19-
zo, |
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za . . . . . . . . . . . .- -. . . . .. .. a J 1 1 1 / 4 * * . , . . . . . . . . . . . . ...0
. . . .. . . ..' . ' / / ( £ ' . . . . . . . . . . . .V .g - - - . . . . . . ' /1 1 1 // /- - - -
YB vj Yd Yb
Figure 13. Example 2: Velocity Vectors in the Plane x = xj
*- .
z
V
Za #W Mw *P -AD -O -1 - - - - - 0. - - -
S 4 I -
n n
B Yj Yd Yb """"-
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Figure 14. Example 2: Velocity Vectors in the Plane x = xf-.--
" * 1. .
-20-
4 1'' . . . . . . . '" ' "
4 4
LI
4 i~4 4 .~•4:. .--- .--
gure 15. Exam ple
2: V locit t I n the
Pans y y.
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421/ " ," " " *"- / / / ,11 "J'I'"
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xi 1o4-f
Figure 16. Example 2: Velocity Vectors In the Plane y Yd
",oS o
-22-
zaz
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Figure 17. Example 2: Pressure Contours In the Plane x xj
-23-
z
za
0 12C-9
Figure 18. Example 2: Pressure Contours In the Plane y = j
-24- *
Xf
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Figure 19. Example 2: Pressure Contours In the Plane z z 2
* L
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-.- 4- 4- 4- .- z- -4 4 4~-h- 9 ~ 9 . 9 ~ - -
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Figure 23. Example 3: Velocity Vectors in the Plane x x I1
za
-------------------------- 4 - - -.- 4 -d -4 -. Ok -4 4- -., -4 -w -F -PI -010
~7 -~ -, -- 7-7 - -q-.p -- pp - - -,- 9 '
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-4-4-4 4 -4- -4-4- -- 416 --. b - 4 4- b - -
-,z -- -9-
YjYdY
Figure 24. Example 3: Velocity Vectors In the Plane x =xf
-28-
Za
Vz
Xi -
Figure\ 25 Exapl 3:Vlct-etr ntePasy
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0-29-
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-30-
z
V
za
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Za
........................
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Figue 2. Exmpl 3:PresureContursIn he Paney y
fx
-32-0
y
47 N
Figure 29. Example 3: Pressure Contours in the Plane z z
y~rXf
*Figure 30. Example 3: Pressure Contours In the Plane z =e
-33-
9 0.
Y\B 2 j YbFigure ~ ~ ~ ~ .31. Exml :xVriiyC mp nn ntePasx
(A%
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Figure 33. Example 4: Velocity Vectors in the Plane x : xj
p. 5 ai a 4 4 i P P ,P Pe P P S L t S. t s. I 4 I • -. .
iti
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, . . . . . . . .,... . -* •4.°. " .° .o.° ° -°o' °° o ' . ° o . . '. - .° °° ,°°°°- - ° . -'° .°o % -" .° - °. "-° % '%". ° •4 -, -. ••
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Figure 36. Example 4: Velocity Vectors In the Plane y Yd
-38-
* S
zaz
z09,V
Ij I I I
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Figue 3 . Ex mpl 4:PresureContursIn he Panex x
1V 1
-39- 0
Zg
Figure 38. Example 4: Pressure Contours In the Plane y =yj
-40-
x
Xf
Sb B
Figure 39. Example 4: Pressure contours In the Plane z= zg
*9
x
(Figure 40. Example 4: Pressure Contours In the Plane z2z
zazy
I I k/ lmit
C 0.
/ : ' od J ~L (J
z -. 1 1
1' I9
Figure~~~~~~~~~~~ 41 Exml :xVriciyCmoetIntePaex
N-.-"
-42-
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Figure 42. Example 4: y-VortlcitY Component In the Plane y -yj
S. . . . . . . . .. . . . . . . .5•... 55. 5 .. 1
z
-43-
I 4 4 4"
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Yj Yd Yb .-
Figure 43. Example 5: Velocity Vectors In the Plans x =xj -.
.
t I t ffI tII Ii 1i &0 p v *- d 54 -0. -
f f fi f I I I 1 % .1 '0'0"9 F it A X . . .9 .0 . 0 - j-9.. .
,,I,.,...........,9999*...*.**.9,
t I l l t P df t * t 4 I P 1 P '0 t4 '0ql ofl '0qq '0q .4 -1 Op Op of Ow X v .0 w . CjoaW
IS "" .9AA0 '
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Figure 46. Example 5: Velocity Vectors In the Plane y myj
-45-
0 z
za - -9 - P
0 P " ,, w ." . 1 t
0~ - w
i* ' ' I 1. *f gls l / / 1 f 4 i t t
' ' 4 / , 1 1 4 4 4 1 4 L t T ' ..* ~ 41 61 4 J 4 4- W4
* d 4" L
Jo.o
v v o i ~ * -w --. 0 : - -, #011 ?
* 9 9 - w -- -0 mo -W *v - -W
-w -w -v - -6 41 -b - 0 - T -#P ~ a
Figure 46. Example 6: Velocity Vectors In the Plane y Yd
........
-46-
L~y
9I I, I oi o I t l I-
Figure 47. Example 5: Pressure Contours In the Plane x x
-47-
z
za
Z9 L
Xi Sf
Fiur 48 x mlp:Pesr onor ntePaey
-48-
y
-Lx
yj ai I
Figure 49. Example 5: Pressure Contours In the Plane z ze
biS-49-
y
rox
* 0j
* f
Figre 0. xaple5: resur Cotous I te Pansz
-50-6
0
ZSS
zgS
Figure 51. Example 5: x-Vortlclty Component in the Plane x =Xj
C-51-
z
*x
ir irr; *
I I I
* a II I* . 'I
I."
I Ici;I I I* a I
I, II I( 1!I a W ~j S
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( * S I. * pI Ia a a * I* a a I* * rI I * I I
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* I I 1I II P.
I P.
I a I a* a
I P.* a a* aI I* I
I -- E ,~ ~ I. a* I* a .j' aI a .1I I *~I' ~'
I 8 d* a Ii./C I a I~***~~%~I I / -~P. P.* S I I.
( I * 1 ~ ~
I I II aa Ia. I
I. A: ~ /' E~',$~S.IE"
2 Ig
XfJ
C
Figure 52. Example 5: y-Vorticlty Component in the Plane y = Yj
C.
*****a.** .. .***-***. ........ .
-52-
yi 53a t5.0 Y
C za
yj53b 10. Yob
C za
Zg
Yj53c t=15.0 Y
Figure 53. Initial Vortex Propagation In Example 5
-53-
54 a y 4
/a is .....
T V
: L9.,,IbI * *1
t 5.5t 11.
16=.5 22=1.0
Fiur 54. Inta otxPoaainIPxml
FILMEDC. Sw
2-85
DTIC
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