The roll-up of a vortex sheet is analyzed by two approaches. The first is based on exact compressible Euler equations while the second is based on the exact incompressible Navier-Stokes equations. The inviscid calculations for the two-dimensional problem do not indicate any roll-up of the sheet. On the other hand, the viscous calculations capture the dynamics of the roll-up rather well. This suggests that the generally held views regarding the roll-up process of aircraft wakes, namely, that it be treated as an inviscid process, may not be completely accurate

Hassan, H. A.

English

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Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19850004563 2020-03-20T19:52:47+00:00Z
(NASA -Ch-1741Oj)
	 PHYSICS CA AlIxChAkl WAKES
	 Ne5-12E71Progress hePort, 15 bay - 14 Nov. 19d4
(North Carolina State Univ.)
	 14 p
dC A021Nk A01
	 C5CL 01A	 UI1C1dL
GJ/02 24629
NASA COOPERATIVE AGREEMENT NCCI-84
PHYSICS OF AIRCRAU WAKES
Progress Report No. 1
For the Period May 15, 1984 to November 14, 1984
Prepared by
H. A. Hassan
Project Coordinator
Department of Mechanical and Aerospace Engineering
P. 0. Box 7910
North Carolina State University
Raleigh, North Carolina 27695-7910
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The progress report consists of the enclosed Abstract which w p -
submitted to the AIAA 18th Fluid Dynamics, Plasma Dynamics and Lasers
Conference.
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Plww.. Cincinnati	 Date. July 16-18, 1985
SeMia►/orgar*w (this information appears in the calf): Dr. Tony Lin
Rw *vderse
11 AIAA has" pubcabon rights to all papas presented at 	 4/ Authors of accepted papers ill be expected to su I it a
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ante d their paper.
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Papa tale (this title wilt be published in the Program):
Roll-up of a Vortex Sheet
Au&mds' name and title. AIAA rmerrcerahip grade, company, full makV address, telephonb number.
1 ALOwprsewwV Path+
	
2
R. A. Mitcheltree, Student Member 	 H. A. Hassan, Associate Fellow
N., C. State University 	 N. C. State University
Box 7910, Raleigh, NC 27695-7910 	 Box 7910, Raleigh, NC 27695-7910
3	 4	 - -
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Summary of Important conclusions:
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Statement of data used to substantiate conclusions, and freehand sketches of major figure-, to be used(no more than two typed pages):
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B28 AIAA BUIlelin/October 1984
ref#	 n
ROLL-UP OF A VORTEX SHEET*
R. A. Mitcheltree**
North Carolina State University, Raleigh, NC
H. A. Hassant
North Carolina State University, Raleigh, NC
Abstract
The roll-up of a vortex sheet is analyzed by two approaches. The first
is bassi on the exact compreF,sible Euler equations while the second is based
on the exact 4ncompressible Na.vier-Stokes equations. The inviscid calcula-
tions for the two-di-iensional problem do not indicate any roll-up of tie
sheet. On the other hand, the viscous calculations capture the dynamics
of the roll-up rather well. This suggests that the generally held views
regarding tale roll-up process of aircraft wakes, namely, that it be treated
as an inviscid process, may not be completely accurate.
Introduction and Avvroach
This research represents the results of the first phase of a research
effort dealing with aircraft wakes. An excellent review (,f this problem
is given by Donaldson and Bilanin. I According to Reference 1, understanding
aircraft wakes entails the understanding of four problems. These are: roll-
up, interaction and stability, aging and atmospheric effects.
*Supported in part by NASA Cooperative Agreement NCCI-84.
**Research Assistant, Mechanical and Aerospace Engineering, Student Member
AIAA.
..
I Professor, Mechanical and Aerospace Engineering, Associate Fellow AIAA
_	 1
2
The object of this investigation is the roll-up region. Its length
scale is of the order of the aspect ratio over the lift cuc ficienc times
the span length. As indicated in Reference'l, processes in rhIs region are
essentially inviscid. Because of this, we felt that an approach based on
the exact Euler equations should refroduce the experimentally observed
features of this region. Such an approach has a distinct advantage over
other approaches reviewed in Faference 1 because the generality of the
equations permits first, detailed study of the relevant dynamics processes
and second, observation of the downstream evolution of vorcical structures
in their proper spatial and temporal relationships.
Al-hough w:? are dealing with low Mach numbers, the compressible form
of the Euler equations is employed. This is a crucial aspect of the form-
k	 ulation because the problem is dominated by boundary conditions, Figure 1.
One does not know before hand whether inflow or outflow conditions should
be imposed at P given boundary. Therefore, without employing a procedure
that selects the proper boundary condition at a given instant and a given
point one cannot hope to achieve an accurate solution. The proper boundary
conditions for the compressible Euler equations, which are hyperbolic, are
determined from the method of characteristics  and these have been imple-
mented here.
The calculation of the development of a vortex wake from a flat sheet,
Figure 2, is considered. This problem, which is characterized as "formid-
able" in Reference 1, is analyzed using the fourth-order Runge-Kutta method
of Jameson et al. 3 Ilse of this method of solution brings up another crucial
aspect of the formulation, namel y , numerical damping. It is extremely
important that the effects of numerical damping be reduced to a minimum in
3order nit to mask the relevant physics of the problem. There are three types
of damping that are employed in Reference 3: second-order, fourth-order
and enthalpy. Because the stagnation enthalpy is not constant for the
problem under consideration, enthalpy damping cannot be used here. Second-
order damping mimics viscous effects and, therefore, was not used. There-
fore, only fourth-order damping was employed in this formulation.
When one employs the compressible equations to study a phenomenon where
the compressibility effects are small, the system of equations becomes stiff.
	
The stiffness is a result of the disparity in the eigenvalues of the system. 	 i
Matrix preconditioning4,5 may be employed to alleviate the problem. There-
fore, a secondary objective of this investigation is to evaluate the utility
of such schemes.
i	 Because of the inability of the Euler equations to predict the physics
of the roll-up problem, the code that was developed in Reference 6 to study
shear layers was employed. The code employs a vorticity-velocity formula-
tion of the incompressible Navier-Stokes equations. Further details of the
method are given in Reference 6.
Results anJ Discussion
The problem considered is that shown in Figure 2. At time t = 0,
the velocity difference between the two streams occurs across one computa-
tional cell with the result that, at t = 0, the vorticity is Zero everywhere
except at that cell. Because enthalpy damping could not be used to speed
up the convergence to the steady state, the matrix preconditioning (M.P.)
procedure of Turkel 5 was employed. Figure 3 shows the convergence history
of the solution based on the Euler equations in the absence and presence of
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matrix preconditioning,. It is ?een from the figure that M.P. helps speed up
the convergence.
Figure 4 shows the vorticity contours that result from calculations
based on the Euler equations with N indicating the number of iterations.
The total number of iterations employed corresponds to the number of
iterations needed to allow a disturbance mooing with the lowest velocity
to propagate across the computational domain. No roll-up is indicated
in the figure. Figure 5 shows the vorticity contours based on the Navier-
Stokes equations. It is seen that the results indicate roll-up.
In conclusion, the above results suggest that the physics of vortex
roll-up is best described by the Navier-Stokes e quations. It appears that
if we hope to understand the behavior of aircraft wakes then we must rely
exclusively on the Navier-Stokes equations.
Refere •.ces
1. Donaldson, C. duP., and Bilanin, A. J., "Vortex Wakes of Conventional
Aircraft," AGARD-AG-204, May 1975.
2. Whitfield, D. L. and Janus, J. M., "Three-Dimensional Unsteady Euler
Equations Solution Using Flux Vector Splitting," AIAA Paper No. 84-
1552, June 1984.
3. Jameson, A., Schmidt, W., ani Turkel, E., "Numerical Solutions of the
Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping
Schemes," AIAA Paper No. 81-1259, June 1981.
4. Briley, W. R., McDonald, H. and Shamroth, S. J., "A Low Mach Number
Euler Formulation and Application to Time-Iterative LBI Schemes,"
ATAA Jou rnal, Vol. 21, October 1983, pp. 1467-1469.
5. Turkel, E., "Accelerat,on to a Steady State for the Euler Equations,"
NASA CR-172398 and ICASE Resort No. 84-32, July 1984.
6. McInvi!le, R. M., Gatski, T. B. and Hassan, }I. A., "Embedded Shear
Layer Computations for Increased Drag R, Juction," AIAA Paper No.
84-0349, 1984.
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Physics on aircraft wakes

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