Richard W, Newsome. Air Force Wright Aeronautical Laboratories. AFSC Liaison Office, NASA Langley Research Center. Bampton, Virginia. James i. Thomas.
Richard W,
Newsome
A i r Force Wright Aeronautical Laboratories AFSC Liaison O f f i c e , NASA Langley Research Center
Bampton, V i r g i n i a James i. Thomas NASA Lanqley Research Center Hampton, V i r g i n i a
The s i m u l a t i o n of t h e leading-edge vortex flow about a s e r i e s of c o n i c a l d e l t a wings through s o l u t i o n of t h e Navier-Stokes and Euler equations is s t u d i e d . The occurrence, t h e v a l i d i t y , and t h e usefulness of s e p a r a t e d flow s o l u t i o n s t o t h e Euler equations a r e of p a r t i c u l a r i n t e r e s t . Central and upwind d i f f e r e n c e s o l u t i o n s t o t h e governing equations a r e compared f o r a series of c r o s s - s e c t i o n a l shapes, i n c l u d i n g both rounded and sharp t i p geometries. For t h e rounded leading edge and t h e f l i g h t condition considered, viscous s o l u t i o n s obtained with e i t h e r c e n t r a l o r upwind d i f f e r e n c e methods p r e d i c t t h e c l a s s i c s t r u c t u r e of v o r t i c a l flow over a highly swept d e l t a wing. Predicted f e a t u r e s include t h e primary v o r t e x due t o leading-edge s e p a r a t i o n and t h e secondary v o r t e x due t o crossflow s e p a r a t i o n . C e n t r a l d i f f e r e n c e s o l u t i o n s t o t h e Euler equations show a marked s e n s i t i v i t y t o g r i d refinement. On a coarse g r i d , t h e flow s e p a r a t e s due t o numerical e r r o r and a primary v o r t e x which resembles t h a t of t h e viscous s o l u t i o n i s p r e d i c t e d . I n c o n t r a s t , t h e upwind d i f f e r e n c e s o l u t i o n s t o t h e Euler equations p r e d i c t a t t a c h e d flow even f o r f i r s t - o r d e r s o l u t i o n s on coarse g r i d s . On a s u f f i c i e n t l y f i n e g r i d , both methods agree c l o s e l y and c o r r e c t l y p r e d i c t a shock-curvature-induced i n v i s c i d s e p a r a t i o n near t h e leeward plane of symmetry. Upwind d i f f e r e n c e s o l u t i o n s t o t h e Navier-Stokes and Euler equations a r e presented f o r two sharp leading-edge geometries. The viscous s o l u t i o n s a r e q u i t e s i m i l a r t o t h e rounded leading-edge r e s u l t s with v o r t i c e s of s i m i l a r shape and s i z e . The upwind Euler s o l u t i o n s p r e d i c t a t t a c h e d flow with no s e p a r a t i o n f o r both geometries. However, with s u f f i c i e n t g r i d refinement near t h e t i p o r through t h e use of more a c c u r a t e s p a t i a l d i f f e r e n c i n g , leading-edge s e p a r a t i o n r e s u l t s . Once t h e leading-edge s e p a r a t i o n i s e s t a b l i s h e d , t h e upwind s o l u t i o n agrees with r e c e n t l y published c e n t r a l d i f f e r e n c e s o l u t i o n s t o t h e Euler equations.
INTIiODUeTION
The c u r r e n t i n t e r e s t i n high angle-of-attack aerodynamics and v o r t i c a l flows has focused considerable a t t e n t i o n on t h e numerical simulation of t h e flow about a swept d e l t a wing a t moderate t o high angles of a t t a c k . For subsonic l e a d i n g edges which a r e sharp o r of small r a d i u s of curvature, t h e flow s e p a r a t e s a t t h e t i p s and f o r m two c o u n t e r - r o t a t i n g v o r t i c e s on o p p o s i t e s i d e s of t h e leeward wing s u r f a c e - The presence of t h e v o r t i c e s produces a p r e s s u r e minimum on t h e upper s u r f a c e and r e s u l t s i n an a d d i t i o n a l l i f t component not p r e d i c t e d by l i n e a r theory.
Interest, here, is restricted to methods whish "capture" the vortex rather than modeling it in an approximate manner. Thus, we consider only methods which solve the Euler and Navier-Stokes equations. The Havier-Stokes equations model all physical mechanisms and provide the most accurate results. Vigneron et al. solved the conical and parabolic approximations to the Navier-Stakes ewations for the vortical flow about a sharp-edged delta wing at supersonic speeds. Fujii and ~utler', solved the three-dimensional Navier-Stokes equations for the leading-edge separation about a delta wing with rounded edges at subsonic speeds. Rizzetta and Shang4 presented threedimensional Navier-Stokes solutions for a delta wing with sharp edges at supersonic and hypersonic speeds. The principal drawbacks of the NavierStokes equations are the higher computational costs necessary to resolve smallscale viscous effects and the need to model turbulence in an approximate manner. However, the Navier-Stokes solutions set the standard by which less exact solutions must be judged. In the last several years, it has been suggested that Euler codes could be the method of choice in the simulation of leading-edge vortex flows. In contrast to potential methods, the Euler equations provide the correct Rankine-Hugoniot shock jump conditions. They also admit rotational flow solutions. Indeed, numerous Euler solutions with leading-edge separation have been reported for both rounded and sharp leading edges using a variety of numerical schemes. A partial list includes the works of Rizzi et a1..6'101 Raj and Sikora, l 1 and Powell, Murman et al. using a finite volume RungeKutta algorithm; Fujii and 0bayashi13 using a LU factored scheme whose righthand side is identical to the Beam and Warmina scheme: and Manie et a1. l 4 and Newsome l using a MacCormack scheme. d
Since flow separation is usually associated with generation of vorticity through the no-slip boundary condition in a viscous flow, its occurrence in an inviscid solution is of both theoretical and practical importance. Necessary conditions for flow separation include the presence of vorticity in the flow as well as an adverse pressure gradient. While the Euler equations admit rotational solutions through the transport (and for three-dimensional flow, stretching) of vorticity, there is only one valid mechanism for vorticity generation in an inviscid flow. In accord with Crocco's theorem, the Euler equations allow for the generation of vorticity through non-constant shock strength (shock curvature, shock intersection, etc. ). Salas16 first demonstrated shock-induced inviscid separation for the transonic flow about a circular cylinder. Marconi17 published similar results for the supersonic flow about circular cones and more recently elliptic cones. l8 The Euler equations are singular at a sharp tip. This, however, causes no particular problem for a finite volume scheme in which cell centered ' ~considering ~ conical flow about quantities are computed. Salas and ~ a ~ w i t tin sharp external axial corners, have shown that a limiting form of the inviscid equations valid at the singular corner point leads to a conical analog of the isentropic Prandtl-Meyer expansion. The maximum Prandtl-Meyer expansion anqle corresponds to vacuum pressure. ~t is not clear whether theoretically valid attached flow Euler solutions exist for geometries in which the vacuum expansion limit is exceeded. For any finite radlus of curvature, the flow field is resolvable and a valid EuLer solution must approach the expansion firnit as the radius of curvature approaches zero. In a viscous gas, the flow separates well before the inviscid expansion limit is reached. Once the
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into the three-dimensional Navier-Stokes equationsr written in terms of the non-dimensional Cartesian variables (x,y,z). Upon simplifying for conical flow, the govering equations may be expressed in conservation form as
A
A
A
The inviscid equations are obtained by dropping the terms (G ,HV,SV). v The general three-dimensional, upwind, Euler/thin layer Navier-Stokes code developed by Thomas2 0 g 2 1 was specialized for conical flow. In the finite volume formulation, a single array of crossflow plane volumes was constructed such that the inflow and outflow surfaces are scaled by the conical transformation, as above. While the code uses a finite volume approach, the equations may be written in generalized coordinates as
At: each iteration, the inflow conditions are updated with the results of the A
previous iteration so that, at convergence, aQ/a( conical flow approximation.
= 0, consistent with the
The inviscid and viscous flux vectors in equations ( 1 ) and (2) are defined as
A
A
A
F,G,H
=
1 J
-
0
a T xxx
+ O S T
xx%P a ~ + X XZ
y x y
S a T
zxz
n
Y YZ
4
"-5
YYY
a
-
+~O S T
Although the flux vectors can fact quite different as applied to dimensional transformation between computational variables is implied be defined as below:
be written in a common form, they are in equations (1) or (2). A general threethe Cartesian variables (x,y,z) and the in equation (2), so that the flux terms can
In the finite volume formulation, expressions for the transformation derivatives and the Jacobain, J, are evaluated geometrically. When working with the conical equations (I), it is convenient to work in terms of the conical variables, Y and Z. This allows a simpler form for the equations using the two-dimensional transformation:
Since
it is convenient to define the terms
so t h a t the flux terms in equation (3) can be defined as
1
The term, - , is absorbed into equation (1) when the Reynolds number and the X time scale are defined with respect to the length scale, L, where L is the length from the body apex to the crossflow solution plane. Upon nondimensionalization in terms of the freestream density, pm and sound speed, cwr the shear stress and heat flux terms are defined in tensor notation (summation convention implied) as:
The chain rule is used to evaluate derivatives with respect to (x,y,z) in terms of (n,
