Optimization of contoured hypersonic scramjet inlets with a least-squares parabolized Navier-Stokes procedure

Computing Systems in Engineering Vol. 4, No. 1, pp. 13-26, 1993

0956-0521/93 $6.00 + 0.00 Pergamon Press Ltd

Printed in Great Britain.

OPTIMIZATION OF CONTOURED HYPERSONIC SCRAM JET INLETS WITH A LEAST-SQUARES PARABOLIZED NAVIER-STOKES PROCEDURE J. J. KORTEt and A. H. AUSLENDER3~ tNASA Langley Research Center, Hampton, VA 23681-0001, U.S.A. :~Lockheed Engineering and Sciences Company, Hampton, VA 23666, U.S.A. Abstract--A new optimization procedure, in which a parabolized Navier-Stokes solver is coupled with

a non-linear least-squares optimization algorithm, is applied to the design of a Mach 14, laminar two-dimensional hypersonic subscale flight inlet with an internal contraction ratio of 15:1 and a length-to-throat half-height ratio of 150: 1. An automated numerical search of multiple geometric wall contours, which are defined by polynomial splines, results in an optimal geometry that yields the maximum total-pressure recovery for the compression process. Optimal inlet geometry is obtained for both inviscid and viscous flows, with the assumption that the gas is either calorically or thermally perfect. The analysis with a calorically perfect gas results in an optimized inviscid inlet design that is defined by two cubic splines and yields a mass-weighted total-pressure recovery of 0.787, which is a 23% improvement compared with the optimized shock-canceled two-ramp inlet design. Similarly, the design procedure obtains the optimized contour for a viscous calorically perfect gas to yield a mass-weighted total-pressure recovery value of 0.749. Additionally, an optimized contour for a viscous thermally perfect gas is obtained to yield a mass-weighted total-pressure recovery value of 0.768. The design methodology incorporates both complex fluid dynamic physics and optimal search techniques without an excessive compromise of computational speed; hence, this methodology is a practical technique that is applicable to optimal inlet design procedures.

NOMENCLATURE

z

#

coefficients of cubic spline specific heat at constant volume internal energy e x and y flux vectors E,F and q flux vectors parabolized ~ flux vector E* least-squares error vector f component of f L enthalpy h axisymmetric source terms H I index of f grid index, axial direction i determinant of (, r/ transformation J Jacobian matrix of f(X) J grid index, y direction J k index kL, kT laminar and turbulent conductivity L inlet length Mach number M m total number of components in f mass flow n total number of components in X total knots on the cubic spline N design objective function Obj pressure P q heat flux gas constant R T temperature time t conservation vector U U, t! x and y velocity components design parameter vector X component of X Xt 3C axial coordinate axial coordinate of spline knots XI N Cartesian or radial coordinate Y wall coordinate Yw ai

Y F 6 A q 0

cv

/-rE,

P O"

7J (O

12

inlet throat half-height ramp angle (relative to the horizontal) grid stretching parameter ratio of specific heats effective gamma, h/e 1, axisymmetric; 0, Cartesian system difference operator crossflow transformation coordinate axisymmetric coordinate laminar and turbulent viscosity streamwise transformation coordinate density safety factor shear stress grid-clustering factor, equal to 0 or 1/2 Vigneron splitting coefficient parabolized source vector

Subscripts t x y 0 ~/

total conditions for e, h, p and T derivative with respect to x derivative with respect to y derivative with respect to 0 derivative with respect to derivative with respect to r/ free-stream condition

INTRODUCTION

Given the restrictions imposed by existing g r o u n d test facilities, the acquisition o f an encompassing hypersonic scramjet data base requires flight experiments. Yet cost and launch capability limitations dictate a significant scale reduction o f single stage to orbit propulsion h a r d w a r e for the developmental tests. Thus, unique subscale-specific design constraints 13

14

J.J. KORTEand A. H. AUSLENDER

exist for each of the three propulsion components: the inlet; the combustor; and the nozzle. Within these constraints, a numerical design methodology is applied to a scramjet subscale flight inlet configuration. Hypersonic full-scale inlet comparison schemes must be efficient and must be achieved within a short length scale because of performance and weight considerations. Ultimately, numerous issues dictate the design of a full-scale flight inlet that is optimally adapted for a unique hypersonic mission scenario. One critical issue is the state of the boundary layer, which is relevant to flow separation physics, heattransfer effects, viscous phenomena, flow-field structure, and inlet performance. For example, the coupling of the boundary layer and full-scale inlet performance for a free-stream Mach number of 16 is significant because the viscous losses generated by the wall boundary layer account for approximately 80% of the net losses incurred during a turbulent hypersonic compression process.1 Other critical issues are the unique test considerations dictated by a reduction in the geometric scale, particularly those that influence boundary-layer transition. Generally, two-dimensional flight scale inlet designs utilize shock cancellation to yield high performance and utilize fully developed turbulent boundary layers to alleviate flow separation problems within the high adverse pressure gradient regions. This dual strategy is not applicable to subscale flight inlet designs because the small physical length scale that is imposed by flight subscale hardware is not compatible with the large physical length scale required for boundary-layer transition. Consequently, two basic approaches for flight subscaleinlet design are frequently proposed. The first approach is to enhance geometrically the laminar-to-turbulent boundary-layer transition by employing an extensive planar-traversing surface that is coupled with a large radial turn before the inlet throat. Unfortunately, this specifies the geometry without regard for the inlet performance. The second approach is to optimize the inlet performance by assuming that the boundary layer never transitions during the compression process. The shortcoming of this approach is the increased sensitivity of the boundary layer to separation, which is an inherent consequence of laminar flow physics. To aid in the analysis of hypersonic inlets, the latter approach is examined with the intent to develop a high-performance laminar subscale flight inlet design methodology that is applicable to realistic experimental flight propulsion hardware. Specifically investigated in this study is the optimization of a subscale scramjet inlet geometry for conditions that are typical of a hypersonic propulsion flight test. The geometry examined is characterized by an internal contraction ratio of 15:1, a fixed length to half-height ratio of 150: 1, and a fixed throat half-height of 2.54 cm (1 in). The propulsion system is proposed to operate at a flight condition of Mach 14 and a dynamic pressure of 95,760 Pa (2000 psf). The integration of the opti-

mal two-dimensional configuration with a viable three-dimensional fright-specific configuration is not considered in this study. This unified inlet design is beyond the scope of this exercise because it requires a simultaneous solution to inlet starting criteria, weight constraints, performance requirements, shock-wave-boundary-layer interaction restrictions, and operability margins. This study is motivated by two subscale inlet flight design issues: the identification of a better design methodology that is less detrimental to performance than a geometrical enhancement of the boundarylayer transition, and identification of a better solution that is obtainable without excessive computational resource allocation. Typically, implementation of a practical inlet design procedure is cumbersome; hence, this study addresses the utility of applying an optimization design method based on computational fluid dynamics (CFD) to simplify, quicken, and broaden inlet design capabilities. Historically, this procedure parallels many analytical approaches to produce optimal aerodynamic designs; however, only a few examples of generalized inlet optimization have been attempted. 2-4Thus, the methodology is a natural extension of prior analytical efforts and utilizes optimization algorithms and efficient CFD algorithms to address problems of hypersonic subscale inlet design. With the advent of modern computer hardware, CFD flow simulations are used increasingly to analyze scramjet flow fields. Numerous CFD algorithms advocated for scramjet inlet analyses are reviewed by White e t al. 5 Optimal CFD-based scramjet component and engine design procedures are outlined by Van Wie e t al. 6 and applied for illustrative purposes to a real-gas Euler analysis of a three-shock ramp inlet configuration. The two-dimensional inlet geometry investigated was comprised of planar wall segments that were defined by angles, which were varied during the optimization procedure. Similarly, this paper investigates two-dimensional inlet geometry and details the optimization of shock-canceled ramp and contoured geometry configurations. Ultimately, the optimal inlet configuration depends on the figure of merit used to define inlet performance. Inlet performance parameters are reviewed extensively by Waltrup e t al., 7 Molder and McGregor 8, and Billig and Van Wie. 9 Numerous methods are presented for characterizing non-idealgas inlet performance. The net entropy production for a steady-state non-ideal adiabatic compression process is readily calculated by employing the totalpressure state conditions. The calculation of the total-pressure state is not used extensively because of the effort necessary to compute the non-ideal-gas state. Specifically, for the nearly adiabatic, laminar shock-dominated compression processes investigated in this study, the mass-weighted total-pressure recovery ratio is utilized as the optimizing figure of merit, which requires the calculation of the air thermodyn-

Contoured hypersonic scramjet inlets amic stagnation state properties presented in Appendix A. This additional analysis is required only at the inlet entrance', and exit and is not a prohibitive computational resource allocation. The computational expense of analyzing the viscous effects with the full Navier-Stokes (NS) equation set is usually prohibitive and restricts the application of' optimization procedures in trying to address realistically the flow physics of complex inlet configurations. Hence, the vast majority of existing CFD-based design studies have been performed on relatively simple geometries that neglects the viscous effects; l°'H however, a least-squares optimization procedure developed recently by Huddleston ~2 was coupled to an efficient parabolized Navier-Stokes (PNS) algorithm 13 by Korte e t al. ~4:5 and applied to a hypersonic wind-tunnel nozzle design. Additionally, for this study, the original PNS CAN-DO ~6computer code was modified to utilize either calorically or thermally perfect thermodynamic gas properties. The application of this PNS design methodology to a purely laminar viscous inlet design is restrictive because transitional or turbulent flow regimes also are readily encompassed by this procedure; however, a comprehensive design methodology that is relevant to flow separation phenomena requires the use of the full Navier-Stokes equation set. Thus, the analysis of the sensitivity of the optimized design to the flow separation effects is not addressed in this study. Rather, the problem that is addressed is the development of an efficient methodology for optimizing the performance of two-dimensional planar and nonplanar inlets. A substantial performance benefit is realized by designs that incorporate curvature, which is demonstrated by comparing the optimal geometry of each design class. The advantage of compressing high-speed flow in as continuous a manner as possible is illustrated. GOVERNING EQUATIONS

In this study, all C F D calculations are made with the CAN-DO code. The governing viscous PNS fluid-dynamic equations and the non-ideal-gas model are described in the subsequent text. The Navier-Stokes equations for two-dimensional Cartesian or axisymmetrie coordinates are U, + E.~ + F, = 6 ( H - - F)/y,

(1)

where U = (p, pu, pv, E = [pu, p u u + p

eT) r

(2)

- "C~x, p u v -- "r~v, u ( e T + p ) - u-C~x -- V-Cxy q-

F = [pv, p u v -- T~,., p v v + p

--

qx]T (3)

r,.:., v ( e T + p )

--ur~,.--w:..,.+q,.] T

(4)

15

H = ( 0 , 0, p -Zoo, 0) T zxx = 2/3(/IL + PT)(2U~ -%,, = 2/3(/~L + #r)(2V,

V,-

(5) fir~y)

-- U~ -- 6 v / y )

(6) (7)

% = (~L+~v)(u,.+v,)

(8)

zoo = 2/3(/~e +/~r) ( - u, - v, + 2r/y)

(9)

q., = - ( k L +

kT)T ~

(10) (11)

q~.= - ( k L + k z ) ~ . ,

where t represents time, x the streamwise coordinate, y the normal two-dimensional or radial (axisymmettic) coordinates and 0 the angular cylindrical coordinate. The density, pressure, temperature, total energy, shear stress, heat flux, molecular viscosity, turbulent eddy viscosity, molecular thermal conductivity, and turbulent eddy thermal conductivity are denoted by p, p, T, eT, ~, q, #L, /tT, kL, and kl, respectively. The total energy is defined by ey = p i e + (u 2 +

t:2)/2],

(12)

where e is the internal energy. For two-dimensional flow, u and v are the streamwise and normal Cartesian velocity components and 3 equals zero. For axisymmetric flow, u and t, are the streamwise and radial velocity components and 6 equals one. Sutherland's law is used to calculate the molecular viscosity. The molecular conductivity is computed with the molecular viscosity and the Prandtl number. The eddy viscosity and eddy thermal conductivity are set to zero for this laminar study. To simulate realistically a scramjet inlet compression process that results in elevated air static temperatures and pressure, high-temperature thermodynamic behavior must be modeled. This complex thermodynamic behavior results principally from the effects of both vibrational excitation and oxygen dissociation. These influences, given a static pressure on the order of 101,325 Pa (1 atm), occur at or above a static temperature of approximately 390 and 1700 K, respectively. These static temperatures and pressures are typically achieved in hypersonic inlet compression processes; the maximum temperature occurs in the boundary layer. Thus, non-ideal thermodynamic gas properties are incorporated into the CFD analysis. The internal energy for an ideal gas is a linear function of temperature, whereas the internal energy e for a thermally perfect gas is a complex function of temperature and is related to p i p with an effective gamma approach: e/(p/p)=F--1.

(13)

16

J.J. KORTE and A. H. AUSLENDER

The effective gamma is defined by

F = h/e

or at both the wall and the centerline. This transformation is given by (14)

and replaces the ratio of specific heats that is commonly employed for ideal-gas calculations. The internal energy is computed by integrating the constant volume specific heat with respect to temperature

e(T) - e(To) =

L

C~(T) dT,

(15)

where cv is defined with a fourth-order polynomial obtained by curve fitting data for air at 1 atm ~7 as

c~/R = 2.7408 - 1.8231(T/1000) + 4.1277(T/1000) 2 -- 2.6068(T/1000) a + (0.55464(T/1000)4(249 K < T < 3000 K) (16a)

cv/R = 5/2 (T < 249 K).

(16b)

yw(x) { y(x,y) = (2~P - -+ 1) fl + 2~P

1 + [(/~+ 1 ) l j flZ 1)11('~-~'>/(1- ~')j} rl = (j -- 1)/(/'max-- 1).

Parabolized Navier-Stokes equations The three-dimensional explicit upwind algorithm presented in Ref. 13 is simplified to two dimensions and modified to include an axisymmetric option. By applying the standard parabolizing assumptions and limiting the transformation to ¢ = ~(x), a fluid dynamic equation results that is comparable with space-marching solution techniques:

J ]¢+

-~

],-

yj

+

\J]~

where

NUMERICAL METHODS To improve numerical efficiency, the PNS equations are solved during the design process. The algorithm used to solve the PNS equations is detailed in Ref. 13.

Equation (1) is transformed into the computational space domain with a generalized transformation (17)

where ¢ represents the streamwise direction and t/ represents the crossflow direction. The transformed NS equations are written as

E* = [pu, puu + cop, puv, u(et + p)]T

(21)

12 = (0, co, 0, 0) T.

(22)

and

Transformation to computational coordinates

r / = t/(x, y),

(19b)

The coefficient ~v is equal to zero for clustering points at the wall and 0.5 for clustering points at both the wall and the centerline. The coefficient fl is a positive number that is greater than unity. The closer/~ is to unity, the more clustered the grid becomes.

This methodology is denoted as a thermally perfect gas because only the temperature is required to specify the air constant volume specific heat.

~ = ~(x,y)

(19a)

The subsonic streamwise pressure gradient is treated with the Vigneron method ~9and the vector f2 is added to eliminate extraneous accelerations/° Vigneron performed a linear stability analysis to determine the value of co required for space-marching stability and concluded that co = min(cr03, 1),

(23)

03 = ?M2/[1 + (7 -- I)MZ] •

(24)

where

The wall radii or heights measured from the centerline yw(X) are defined with a set of points. For a given streamwise location, Roberts' stretching transformation 18 is used to duster points either at only the wall

In this investigation, 03 is applied with a safety factor cr that ranges from 0.65 to 0.95. A two-stage explicit upwind scheme is used to integrate the equation set in the streamwise direction. Second- and third-order upwind flux gradient approximations are employed to integrate the PNS equations and are constructed with a modified Roe

Contoured hypersonic scramjet inlets

15 ~

17

~-

10 y/z

O~2

~' , , ,

0

25

50

75 x/z

_ ,

,

~

.

~zone 4

,

1O0

125

150

Fig. 1. Nomenclature for simple shock-canceled two-ramp inlet system• methodology. Additionally, the numerical fluxes are treated at local extrema to eliminate spurious oscillations with a conventional minmod limiter. NON-LINEAR LEAST-SQUARES OPTIMIZATION

This study utilizes the least-squares parabolized Navier-Stokes (LS/PNS) optimization procedure discussed in Ref. 15. For clarity, a brief summary is presented in the following section. Objective function

A non-linear optimization problem is generally solved by minimization of an objective function. The objective function Obj is implicitly dependent on a set of design parameters X and is constructed from a series of functions f / o f the non-linear least-squares f o r m 21

Obj(X) = ~ f~(X).

Like the objective function, design parameter selection is unique to each problem; an ideal set is strongly coupled to the objective function and contains exactly the minimum number of required elements. For the design of a scramjet inlet, the inflow conditions are specified and the wall contour is undetermined; therefore, in this study, the design parameters are a set of coefficients used to define analytically the wall boundary. Wall contour

This investigation utilizes piecewise spline contours to characterize the wall geometry and a consistent set of design parameters that vary in number dependent upon the degree of flexibility desired. Specifically, the entire inlet contour is defined with N cubic polynomial equations; each equation denotes one local spline contour;

(25) O
l=1

•,

f,,,]T

(26) XN_ I
X = [X,, X2, . . , X,,]T

(27)

and the objective function is expressed as Obj(X) = fr(x)f(X).

(28)

For each minimization problem, a number of relevant definitions exist for f; however, f should be defined in a manner that forces the objective function to exhibit a strong minimum. The components of f chosen for this study are defined by the mass-weighted totalpressure losses at the inlet throat J / = P, ,ui4AYij(P,,j -Pt~, )/(mpt~).

{ y , , = a o + a l x 4-a:x24-a3x 3

xl < x <~x2 {y,~=a 4 + a S x 4-a6x 2 + a 7 x 3

Two vectors f and X are defined as f= ~,f2,.

Design parameters

(29)

The i, j index :identifies the grid point at the exit of the inlet and p,~ is the total pressure at the entrance of the inlet. For brevity, mass-weighted total-pressure recovery is referred to throughout the remainder of this text as simply total-pressure recovery.

~X N

{Yw= a4N-4 4- a4N

+a4N l x3-

3X

4- a4, ~,

~.2

2-

(30)

The N cubic equations result in 4N coefficients, with an assumption that the locations xl through x~ are known. The requirement of continuity of the surface, slope and curvature at x I through x~. specifies 3 ( N - 1) of the coefficients. The remaining N 4-3 coefficients are solved for with the known inlet and exit radii and slopes, as well as by requiring the specification of the slopes at other N - 1 locations• After the coefficients are defined, a linear system of equations is solved to determine the value of the coefficients ao - a4N_ I- TO model a ramp inlet, which is a degenerate case, the coefficients associated with x 2 and x 3 are set to zero. This procedure allows the inlet geometry to be detailed without an excessive number of parameters and additionally yields geometry with continuous first derivatives, which is an important factor in designing high-performance supersonic compression systems.

J. J. KORTEand A. H. AUSLENDER

18

Table 1. Simple shock-canceled two-ramp inlet system parameters for calorically perfect gas with a contraction ratio of 15:1 and L/z = 150 Zone 1 2 3 4

Flow angle (deg) 0.000 2.882 9.342 0.000

Mach 14.000 12.106 8.953 6.385

Temperature (K) 233 309 550 1023

Non-linear optimization by solving a least-squares problem

RESULTS AND DISCUSSION

A series of two-dimensional inlet compression schemes are investigated, each constrained to a fixed contraction ratio, a fixed length and, if viscously analyzed, to a fixed wall temperature of 556K (1000°R). For this investigation, two-dimensional inlet wall geometry is defined by employing either linear segments or cubic splines and is referred to as a ramp or a contoured inlet, respectively. Six cases are considered that encompass increasingly more complex geometry and fluid dynamic assumptions. For the inviscidly analyzed, simple shock-canceled tworamp inlet configurations, the C F D calculations and the resultant total-pressure recoveries are compared with analytical solutions. For the calorically perfect gas cases, the total-pressure recovery is computed with ideal-gas relationships. For the thermally perfect gas cases, total-pressure recovery is determined with the non-ideal partition-function thermodynamic state calculation detailed in Appendix A. Additionally, numerous inlet designs are generated, which employ calorically and thermally perfect gases, for inviscid optimized shock-canceled ramp inlets and for inviscid and laminar viscous optimized contoured inlets.

(31)

where the design parameters are explicitly updated as Xk + 1 = Xk + AXk.

1.000 0.9269 0.6201 0.3884

adequate results when computed by specifying AX~ to be X~ x 10 -5.

Recall that Obj(X) has a local extremum that coincides with the local gradient of zero. The small residual form of Newton's method for non-linear least-squares is utilized as J~JkAXk = - J~fk,

P~/Pt~

7 1.4 1.4 1.4 1.4

(32)

For additional information on methods for solving Eq. (31) for both small and large residual problems, see the work of Scales. 2t

Jacobian matrix The numerical construction of the Jacobian matrix is the most computationally intensive part of the optimization algorithm because each component requires one additional flow-field solution. The elements of the Jacobian matrix, which are derivatives, are approximated with central finite differences. The accuracy of the difference approximation is dependent on both the magnitude of AXs and the non-linearity of the flow-field solution and yields 15

-10

--15

'

0

I

[

25

I

I

I

I

I

50

I

I

I

'

I

J

75

=

i

l

I

100

i

i

i

i

I

125

i

t

i

t

I

150

x/z

Fig. 2. Comparison of Mach-number contours for the analytic and CFD solutions of the simple shock-canceled two-ramp inlet (calorically perfect gas, ? = 1.4, inviscid analysis).

Contoured hypersonic scramjet inlets

19

Table 2. Simple non-ideal shock-canceled two-ramp inlet system parameters with a contraction ratio of 15:1 and L/z = 150 Zone

Mach

Flow angle (deg)

Temperature (K)

7

Pt/Pt~

1 2 3 4

14.000 12.085 9.208 6.869

0.000 2.931 9.019 0.000

233 311 527 929

1.400 1,398 1,382 1.340

1.000 0.9264 0.6628 0.4270

in this study, the Vigneron approximation is not invoked. Thus, the algorithm solves the Euler equations, except for the negligible influences of the viscous terms included in the stress tensor. For the CRAY-YMP, typical expended CPU times are 10 s for an inviscid flow solution and 130s for a viscous flow solution. Contour plots are constructed with data saved at 400 equally spaced streamwise stations.

These results validate the design procedure and are detailed in the subsequent text. The CFD calculations are made on a CRAYYMP. The inflow boundary conditions are consistent with a free-stream flight condition of Mach 14 at an elevation of 33,650 m (110,400 ft) and correspond to a static temperature of 233K (419°R) and a free-stream Reynolds number of 2.95× 106m -l (74,900 in-1). For viscous calculations, the wall temperature is assumed to be 556K (1000°R). The constrained geometric parameters of the system are: a contraction ratio of 15: 1; an inlet throat half-height of 2.54cm (lin); and an inlet length of 381cm (150in). A constant-area section with a length of 25.4 cm (10 in) is located downstream of the throat and is employed to aid in the assessment of inlet performance. For each specified inlet configuration, the total-pressure recovery is calculated at the exit of this section. The numerical grid in the normal direction is constructed with 63 points and is clustered at the wall and centerline boundaries with a gridstretching factor/3 = 1.04. All results are computed with a third-order inviscid upwind flux differencing stencil. The numerical grid in the streamwise direction numbers between 10,000 and 160,000 stations, depending on the type of wall boundary conditions (slip or no slip) imposed. The slip wall boundary condition is used in the PNS calculations to obtain an inviscid flow-field solution. For the inviscid cases

Case I: simple shock-canceled two-ramp inlet, calorically perfect gas, and inviscid flow A simple shock-canceled two-ramp inlet system (Fig. 1) is investigated both analytically and computationally to test the code and validate the calculation of total-pressure losses. An exact, four-zone analytic solution is obtained with oblique shock-wave analysis. The resulting geometry and associated flow solution for each zone are given in Table 1. The CFD solution for the identical geometry is computed by using the PNS flow solver with a slip wall boundary condition. The Mach-number contours for the analytic and CFD solutions are shown in Fig. 2. The clustered Mach-number contours delineate the shock-wave locations of the CFD solution. As shown in Fig. 2, good agreement is obtained between the analytical and computational results. A total-pressure recovery of 0.393 is calculated with the CFD solution and compares favorably with the analytical solution of 0.388, which yields a difference of only 1.29%. The

15

10

-10

--15

~

0

L

I

25

i

i

i

i

I

50

i

i

i

~

I

~

75

i

i

i

I

100

i

i

i

i

I

125

i

i

I

I

'1

150

x/z

Fig. 3. Comparison of Mach-number contours for the analytic and CFD solutions of the simple shock-canceled two-ramp inlet (thermally perfect gas, inviscid analysis).

20

J . J . KORTE and A. H. AUSLENDER

10.0

0.64 0.6

9.0

0.56 0.52 0.48

8.0

0.44

%, deg

0.4

7.0

0.36 0.32 0.28

6.0

0.24 3.0

3.5

4.0

(X1,

4.5

5.0

deg

Fig. 4. Two-ramp inlet total-pressure recovery variation as a function of compression surface angles (calorically perfect gas, 7 = 1.4, inviscid analysis).

14.1 13.3 12.5 11.7 10.9 10.1 9.3 8.5 7.7 6.9 6.1 0

0~1 ---- 4 . 2 6 3 *

15

10

5

y/z o -5

-10 Total-Pressure Recovery = 0.625 m15 0

25

50

75

100

125

150

x/z Fig. 5. Computed Mach-number contours for optimized two-ramp inlet (calorically perfect gas, y = 1.4, inviscid analysis).

Contoured hypersonic scramjet inlets 1

5

21

~

10 y/z

Design parameters

~

/

X(3) = dYw(X2) / dx i

0

,

,

i

I

25

i

i

i

h

I

J

50

i

" ~ S e c o n d spline i

,

i

.

.

.

.

75

i

.

.

.

.

1O0

i

.

125

.

.

.

I

,

I

Z

150

x/z

Fig. 6. Nomenclature for contoured inlet system with two piecewise cubic splines. difference in total-pressure recovery between the C F D and analytical solutions is examined though a grid resolution study. The C F D code is run repeatedly with an increasing number of grid points (40, 63, 93, 140, and 210) in the direction normal to the centerline, which results in total-pressure errors of 2.29, 1.29, 0.88, 0.52, and 0.28%, respectively. For this study, an error of approximately 1.00% is deemed acceptable; hence, all C F D calculations are generated with a normal grid comprised of 63 points.

Case H: simple shock-canceled two-ramp inlet, thermally perfect gas, and inviscid flow To test the'. predictive ability of the thermally perfect gas code, a simple shock-canceled two-ramp inlet is analyzed. An exact, four-zone analytic sol-

ution is obtained with oblique shock-wave analysis and non-ideal thermodynamic state calculations. The resulting geometry and associated exact flow solution for each zone are given in Table 2. Non-ideal thermodynamic effects result in an approximately 10% variation between the total-pressure recoveries of the analytic solutions of case I and case II. Also, the two distinct configurations yield different exit Mach numbers. The C F D solution, which employs the geometry specified in Table 2, is computed by using the thermally perfect PNS flow solver with a slip wall boundary condition. The Mach-number contours for the analytic and C F D solutions are shown in Fig. 3. The regions delineated by the Mach-number contours illustrate that the shock structure compares well with that of the analytical solution. A total-pressure recov-

14.1 13.3 12.5 11.7 10.9 10.1 9.3 8.5 7.7 6.9 6.1 0

15

10

5

y/z

o -5

-10 Tolal-Pressure Recovery = 0.787 -15 0

25

50

75

100

125

150

xlz Fig. 7. Computed Mach-number contours for contoured inlet system with two piecewise cubic splines (calorically perfect gas, ~' = 1.4, inviscid analysis).

22

J.J. KORTEand A. H. AUSLENDER

14.1 13.3 12.5 11.7 10.9 10.1 9,3 8.5 7.7 6.9 6.1 0

15

10

y/z

o -5

-10

-15

0

25

50

75

100

125

150

x/z Fig. 8. Computed Mach-number contours for contoured inlet system with two piecewise cubic splines (calorically perfect gas, 7 = 1.4, viscous analysis). cry of 0.439 is calculated with the CFD solution and compares favorably with the analytic solution of 0.427, which yields a difference of 2.8%. The deviation is partially caused by the fact that the thermodynamic air curve fit employed by the CFD code is not a function of static pressure, which results from the compromise between the requirement for computational efficiency and accuracy. This error is deemed acceptable for this study.

Case III: optimum shock-canceled two-ramp inlet, calorically perfect gas, and inviscid flow The simple shock-canceled inlet is not the best configuration for maximum total-pressure recovery. The LS/PNS procedure is used to determine a new two-ramp inlet design for improving total-pressure recovery. The two ramp angles, ~1 and c~2, are specified as the design parameters. To investigate the behavior of the total-pressure recovery as a function of the two design parameters, a matrix of 29 x 29 runs is computed. The variation of total-pressure recovery as a function of these two parameters is shown in Fig. 4. A maximum value of the totalpressure recovery is observed to be approximately 0.640. The optimum ramp angles are approximately ~q = 4 ° and ~2 = 8°. This result is consistent with the classical optimum inlet design criterion, which requires compression schemes that generate waves of

nearly equal strength. Thus, for the optimum tworamp inlet, the second ramp angle is approximately two times larger than the first ramp angle. The initial geometry used to start the optimum procedure is given in Table 1. The total-pressure recovery improves from 0.390 for the initial geometry to over 0.600 in two iterations; however, the LS/PNS procedure experiences difficulty in obtaining a converged optimum solution and varies between 0.600 and 0.625 for additional iterations. The Machnumber contours for the improved design are shown in Fig. 5. The resultant compression scheme still exhibits shock-canceling at the inlet throat, but the flow structure is more complex. The large inefficient supersonic turning angles of the simple shockcanceled ramp inlet are segmented into more efficient, smaller turns to yield a net gain in inlet performance.

Case IV: optimized contoured &let, calorically perfect gas, and invisicid flow The optimum shock-canceled two-ramp inlet is not the best design for maximizing total-pressure recovery. Unfortunately, an ideal isentropic compression process is not possible because of the constrained inlet length; however, a nearly isentropic compression scheme is possible after the flow is processed by the leading-edge shock wave if the inlet is properly contoured. To investigate this, a contoured inlet is

Contoured hypersonic scramjet inlets

23 14.1 13.3 12.5 11.7 10.9 10.1 9.3 8.5 7.7 6.9 6.1 0

15

10

5

y/z

o -5

-10

-15 0

25

50

75

100

125

150

x/z Fig. 9. Computed Mach-number contours for contoured inlet system with two piecewise cubic splines (thermally perfect gas, viscous analysis). designed with two cubic splines. The design problem is specified by three design parameters that consist of the starting slope [dyw(O)/dx] of the first spline and the starting location (x2) and slope (dy,(xz)/dx] of the second spline (Fig. 6). The slope at the inlet throat is continuous and set to zero. To prevent non-physical wall geometry, the starting location of the second cubic spline is constrained to be less than that of the inlet throat location. The length, capture area, and contraction ratio of the inlet design are held constant. The Mach-number contours for the optimized geometry are shown in Fig. 7. The reflected shock wave that emanates from the centerline is processed by a series of compression waves that yield a nearly shock-canceled wave structure at the throat, which results in a total-pressure recovery of 0.787. This recovery represents a 23% increase compared with the optimum two-shock ramp inlet configuration examined in case III. To obtain an inviscid shock-canceled inlet compression scheme, the slope of the throat must be discontinuous; however, this investigation specifies a Table 3. Design parameters and total-pressure recovery for optimized curved inlets Case IV V VI

dyw(O)/dx -0.0783 -0.0666 -0.0633

x2/z 149.74 141.26 140.93

dyw(x2)/dx

Pt/P~

-0.1817 -0.1794 -0.1776

0.787 0.749 0.768

continuous derivative for the entire inlet contour. Hence, in an attempt to approximate the discontinuous optimal throat geometry, the design procedure reduces the second cubic spline to a minimum length and simultaneously increases the local curvature which effectively yields a geometry that nearly satisfies the optimal shape requirements and does not violate the problem constraints. The specification of a continuous derivative constraint is utilized to address viscous effects because geometric discontinuities promote flow separation.

Case V: optimized contoured inlet, calorically perfect gas, and laminar flow A realistic inlet design must consider boundarylayer influences on performance; therefore, the previous case is reexamined with the assumption of laminar flow. The optimized inviscid design is used to restart the optimization procedure. The laminar viscous influences on the optimized inviscid design are small and reduce the total-pressure recovery by 6.2% to 0.738. The final, optimized viscous design improves the total-pressure recovery to 0.749. The Mach-number contours for the optimized geometry (Fig. 8) have a similar flow structure when compared with the computed Mach-number contours for the inviscid configuration (Fig. 7), yet the design parameters are slightly varied. The initial slope of the first spline is decreased in magnitude to account for

24

J.J. KORTEand A. H. AUSLENDER

the laminar boundary-layer growth. Also, the starting location of the second spline moves closer to the origin compared with the inviscid design, which allows the throat region to accommodate the boundary layer. Case VI: optimized contoured inlet, thermally perfect gas, and laminar flow A realistic inlet design must consider the effect of non-ideal gas behavior on performance. The optimized, calorically perfect design for laminar flow is used to restart the optimization procedure. This results in a total-pressure recovery of 0.764; further application of the optimization procedure results in a small improvement to 0.768. The Mach-number contours for the optimized geometry shown in Fig. 9 have a flow structure similar to that of the previous contoured inlet design. The results for case VI are compared with the other contoured inlet designs in Table 3. CONCLUDING REMARKS

A CFD-based optimization procedure that utilizes a least-squares minimization technique coupled to a parabolized Navier-Stokes solver is applied to the design of two-dimensional hypersonic inlets. The figure of merit applied in the optimization procedure is the total-pressure recovery at the exit of the inlet system. A specific hypersonic flight condition and subscale configuration with a number of Mach 14, an internal contraction ratio of 15:1, and a length-tothroat half-height ratio of 150:1 is analyzed with inviscid and viscous flow. This design procedure is validated with analytic inviscid test cases for simple shock-canceled geometry. Two thermodynamic gas models (calorically and thermally perfect) are employed in this study. For the thermally perfect gas CFD cases, total-pressure recovery is determined with a non-ideal partition-function thermodynamic state calculation. The CFD-generated total-pressure values for the two validation cases agree with the analytical solution to within 1.3 and 2.8%, respectively, for the calorically and thermally perfect gas assumptions. Significant inlet performance improvements are obtained for both two-ramp and contoured inlet systems. The optimum inviscid two-ramp design for a calorically perfect gas obtained a total-pressure recovery of 0.625. A contoured inlet design is constructed with two cubic splines. The optimum inviscid contoured inlet for a calorically perfect gas obtained a total-pressure recovery of 0.787. Optimum viscous contoured inlet designs are computed for the calorically and thermally perfect gas assumptions. These designs yield a total-pressure recovery of 0.749 and 0.764 for calorically and thermally perfect gas assumptions, respectively. The rapid evaluation of inlet configurations with increasing complex geometry and flow physics is

demonstrated. This automated numerical optimization study also shows the utility of incorporating curvature in the inlet geometry to enhance the totalpressure recovery. In general, this methodology is envisioned for application to more complex problems. Future investigations will address two-dimensional and three-dimensional inlet geometry and performance objective functions consistent with flow separation criteria, which will enhance the LS/PNS procedure as a more comprehensive hypersonic inlet design tool.

REFERENCES

1. C. Y. Anderson, "Hypervelocity diffusive burning scramjets," Proceedings of the Performance Enhancement For Hypervelocity Airbreathing Propulsion Workshop, NASA Langley Research Center, Hampton, Virginia, 29-31 July 1991, p. 38. 2. K. Oswatitsch, "Pressure recovery for missies with reaction propulsion at high supersonic speeds (the efficiency of shock diffusers)," NACA TM-1140, June 1947. 3. D. M. Van Wie, M. E. White and P. J. Waltrup, "Application of computational design techniques in the development of scramjet engines," AIAA Paper No. 87-1420, June 1987. 4. J. F. Connors and R. C. Meyer, "Design criteria for axisymmetric and two-dimensional supersonic inlets and exits," NACA TN-3589, January 1956. 5. M. E. White, J. P. Drummond and A. Kumar, "Evolution and application of computational design techniques in the development of scramjet engines," AIAA Paper No. 87-1420, June 1987. 6. D. M. Van Wie, M. E. White and P. J. Waltrup, "Applications of computational design techniques in the development of scramjet engines," AIAA Paper No. 87-1420, June 1987. 7. PI J. Waltrup, F. S. Billig and R. D. Stockbridge, "A procedure for optimizing the design of scramjet engines," Journal of Spacecraft and Rockets 16, 163-172 (1979); also AIAA paper No. 78-I110, July 1978. 8. S. Molder and R. J. McGregor, "Analysis and optimization of scramjet inlet performance," Proceedings of the 17th ICAS Congress, Stockholm, Sweden, 2, 1990, pp. 1328-1339. 9. F. S. Billig and D. M. Van Wie, "Efficiencyparameters for inlets operating at hypersonic speeds," ISABE 877047. 10. O. Baysal and M. E. Eleshaky, "Aerodynamic design optimization using sensitivity analysis and computational fluid dynamics," AIAA Journal 30, 718-725 (1992). 1I. P. D. McQuade, S. Eberhardt and E. Livne, "Optimization of a 2D scramjet-vehicleusing CFD and simplified approximate flow analysis techniques," AIAA Paper No. 92-3673, July 1992. 12. D. H. Huddleston, "Aerodynamic Design Optimization Using Computational Fluid Dynamics," Ph.D. dissertation, University of Tennessee, Knoxville, 1989. 13. J. J. Korte, "An explicit upwind algorithm for solving the parabolized Navier Stokes equations," NASA TP 3050, February 1991. 14. J. J. Korte, A. Kumar, D. J. Singh and B. Grossman, "Least squares/parabolized Navie~Stokes procedure for optimizing hypersonic wind-tunnel nozzles," AIAA Journal of Propulsion and Power 8, 1057-1063 (1992); also AIAA Paper No. 91-2273 June 1991. 15. J. J. Korte, "Aerodynamic design of axisymmetric hypersonic wind-tunnel nozzles using least-squares/

25

Contoured hypersonic scramjet inlets

16.

17. 18. 19.

20.

21. 22. 23.

24.

25.

26.

parabolized Navie~Stokes procedure," AIAA Journal o f Spacecrqft and Rockets 29, 685~i91 (1992); also AIAA Paper No. 92-0332, January 1992. J. J. Korte, A. Kumar, D. J. Singh and J. A. White, "C_AN-DO-(2FD-based Aerodynamic Nozzle Design & Optimization program for supersonic/hypersonic wind tunnels," AIAA Paper No. 92-4009, July 1992. W. M. Rohsenow and J. P. Hartnett, Handbook o f Heat Transfer, pp. 91 92, McGraw-Hill, New York, 1973. D. A. Anderson, J. C. Tannehill and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, pp. 247-251, McGraw-Hill, New York, 1984. Y. C. Vigneron, J. V. Rakich and J. C. Tannehill, "'Calculation of supersonic viscous flows over delta wings with sharp subsonic leading edges," AIAA Paper No. 78-1137, July 1978. J. H. Morrison and J. J. Korte, "Implementation of Vigneron's streamwise pressure gradient approximation in the PNS equations," AIAA Paper No. 92-0189, January 1992. L. E. Scales, Introduction to Non-Linear Optimization, pp. 110 136, Macmillan, London, 1985. G. N. Lewis and M. Randell, Thermodynamics, 2nd edn, pp. 419~447, McGraw-Hill, New York, 1961. J. O. Hirschfelder, C. F. Curtis and R. B. Bird, Molecular Theory of Gases and Liquids, p. 117, John Wiley, New York, 1964. V. N. Huff, S. Gordon and V. E. Morrell, "General method and thermodynamic tables for computation of equilibrium composition and temperature of chemical reactions," NACA Report 1037, 1951. C. E. Moore, "Atomic energy levels as derived from the analyses of optical spectra," Vol. 1, Circular of the National Bureau of Standards 467, 1949. J. Hilsenrath and M. Klein, "Tables of thermodynamics properties of air in chemical equilibrium including second virial corrections from 1500 to 15,000K," AEDC-TDR-63-161, August 1963.

constant, #, is the molecular weight of the ith species, T is the system temperature, p,~tmis the partial pressure of the ith species (measured in units of atmospheres), and LYJis t h e j t h energy state of the partition function for the lth energy mode. The functional relationship for the internal modes is explicitly detailed. Consistent with prior analysis, the translational mode is quantified and yields the standard Sackur-Tetrode equation.23 With manipulation, this equation set yields the Gibbs free energy for an individual constituent. G,= H , - TS, = f o + R T ln(P~ tin)

{

(A3)

,

f o = R T 3.6651 -321n(~,)- ; l n ( r ) _ M~°ss ~ s [ln(2J)ll + H V..... ,,n, ) /=1 j=l

(A4)

where G~ is the Gibbs free energy of the ith constituent and f 0 is the Gibbs free energy of the ith constituent at the reference pressure, and is easily extended to an arbitrary mixture Species

Gui ..... = ~

Species

n,G,= ~_, n,[f,'+RTln(P',"m)],

~-I

i

1

(A5) where n~ is the number of moles of the ith The air equilibrium molar composition system pressure and system temperature is numerical Gibbs free energy minimization yield a solution to

F

species. at a specified derived with a procedure 24 to

//s~.,~ + R T ln(P arm) Elements

A P P E N D I X A: H I G H - T E M P E R A T U R E

+ Ek=

AIR M O D E L

The high-temperature equilibrium air model is a direct application of classical Bose-Einstein statistical thermodynamics, which utilizes multiple constituents: diatomic oxygen, atomic oxygen, ionized oxygen, diatomic nitrogen, atomic nitrogen, ionized nitrogen, nitrous oxide, and an electron. The partition fimctions 22 for the species incorporate the modeling of fi)ur distinct energy modes: translational, rotational, vibrational, and electronic. Also included in the partition function are the standard rotational and vibrational correction terms that are required for accurate thermodynamic predictions. In the model, only the translational partition function is volumetrically dependent. In addition, this function is volumetrically dependent to only the first power, so that the resultant equilibrium system satisfies the ideal equation of state. The subsequent equations are employed for the characterization of a constituent entropy state and a constituent enthalpy state on a per mole basis: S, R = [~ In(#,) ÷ ~ ln(T) -- ln(P~''m) -- 1.165l 1

+

~ [ln(2~) + r(? T ln(2~) k /=l

(A1)

1=1 L

2ka,.k = 0

(A6)

I

Species

~, i

niai,k =

mk,

(A7)

I

where a~.k is the number of k elemental atoms in the ith species, A k is the number of k elemental atoms in the system, p,tm is the system pressure (measured in units of atmospheres), and 2k is the Lagrange parameter of the k th element. The mixture is denoted in the body of the paper as calorically perfect if: (1) the Gibbs free energy minimization calculation is not utilized; (2) the molar composition is constant for all temperatures and pressures; and (3) the species partition functions only account for translational and rotational energy modes. If the Gibbs free energy minimization calculation is not circumvented and all energy modes are employed, then the equilibrium mixture is denoted in the body of the paper as a non-ideal gas. Note that the pertinent issue for the high-temperature equilibrium air modeling is the construction of valid electronic partition functions for quantifying the numerous constituent Gibbs free energy functions defined at the reference pressure (1 atm). This is accomplished by the direct employment of species spectral data 25 to construct the relevant internal energy partition functions as States

2Et...... i¢= ~ g/e e, kf,

(A8)

/-I

RT

-

2 + ,--t j~-L h T ~ T l n ( 2 J )

,

(a2)

where S, is the entropy of the ith constituent, H~ is the enthalpy of the ith constituent, H I °~'~°" is the heat of formation of the ith constituent, R is the universal gas

where ~E~c,ro,,¢ is the electronic energy partition function, gj is the degeneracy of t h e j t h electronic energy state, E~ is the energy of the j t h electronic state (obtained from spectra data) and k is Boltzmann's constant. This provides a simple closure procedure. The resulting high-temperature air equi-

26

J . J . KORTE and A. H. AUSLENDER

librium thermodynamic algorithm is applicable to hypervelocity stagnation states and is in good agreement with other existing high-temperature air methodologies such as that of Hilsenrath and Klein. 26 This equation set establishes all equilibrium thermodynamic properties as explicit functions of pressure and temperature. Thus, if variables other than pressure and temperature are parameters, then an additional numerical iteration procedure is required to solve the non-linear

system. This procedure is accomplished by employing a bounded Newton-Raphson iteration technique; therefore, any two state variables are admissible as the independent thermodynamic coordinates. Generally, for the calculations of this investigation, the inlet equilibrium stagnation state is characterized by the thermodynamic variables of total enthalpy and entropy, and the associated total pressure and total temperature are deduced iteratively.