numerical study of base bleed effects on aerodynamic drag for a transonic projectile

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [HEAL-Link Consortium]On: 14 April 2011Access details: Access Details: [subscription number 786636650]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Computational Fluid DynamicsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713455064

NUMERICAL STUDY OF BASE BLEED EFFECTS ON AERODYNAMICDRAG FOR A TRANSONIC PROJECTILEJan-Kaung Fua; Shen-Min Liangb

a Department of Aeronautics, Chinese Air Force Academy, Kangshan, Kaohsiung, Taiwan b Institute ofAeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan

To cite this Article Fu, Jan-Kaung and Liang, Shen-Min(1993) 'NUMERICAL STUDY OF BASE BLEED EFFECTS ONAERODYNAMIC DRAG FOR A TRANSONIC PROJECTILE', International Journal of Computational Fluid Dynamics, 1:3, 249 — 273To link to this Article: DOI: 10.1080/10618569308904475URL: http://dx.doi.org/10.1080/10618569308904475

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713455064

http://dx.doi.org/10.1080/10618569308904475

http://www.informaworld.com/terms-and-conditions-of-access.pdf

Comp. Fluid Dyn., 1993, Vol. I , p. 249-273 O 1993 Gordon and Breach Science Publishers, S.A. Reprints available directly from the publisher Printed in the United States of America Photocopying permitted by license only

NUMERICAL STUDY OF BASE BLEED EFFECTS ON AERODYNAMIC DRAG FOR A TRANSONIC

PROJECTILE

JAN-KAUNG FU

Department of Aeronautics, Chinese Air Force Academy, Kangshan, Kaohsiung, Taiwan 82012, R.O.C.

SHEN-MIN LIANG

Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, R.O.C.

(Received 10 September 1992; in Jinaljorm 16 April 1993)

ABSTRACT

A numerical study is made to analyze the performance of a secant-ogive-cylinder projectile in the transonic regime in terms of aerodynamic drag. At transonic speeds, the base drag contributes a major portion of the total aerodynamic drag, and hence affects projectile's performances significantly. The base bleed method is applied to reduce the base drag by varying the value of parameters, the bleed quantity ( I ) and the bleed area ratio (G). The implicit, diagonalized, symmetric Total Variation Diminishing (TVD) scheme, accompanied by a suitable grid, is employed to solve the thin-layer axisymmetric Navier-Stokes equations coupled with the Baldwin-Lomax turbulence model. The computed results show that, in comparison with the case without base bleed, an increase in bleed quantity or a higher injection speed due to a smaller bleed area ratio at fixed bleed quantity can result in a base (and total) drag reduction. At Mach number 0.96, the reductions in base drag and total drag can be as high as 64% and 44%, respectively, for I= 0.1 and 0 = 0.3.

1. INTRODUCTION

The secant-ogive-cylinder (SOC) configuration, as shown in Figure 1, has become one of the standard models for the artillery projectile and/or the missile shell design'.'. The critical aerodynamic behaviour of projectiles in the transonic regime can be attributed in part to the complex shock structure existing on the projectile. Since axisymmetric bodies, like missiles, artillery projectiles and bullets,.often have a blunt base, the corresponding base drag is usually an appreciable part (typically about 50%)' .~ of the total aerodynamic drag for the transonic speed flight, and significantly influences the projectiles performances such as the flight range and the terminal velocity. The pressure and viscous components of aerodynamic drag generally cannot be reduced significantly without adversely affecting the stability of

Downloaded By: [HEAL-Link Consortium] At: 04:35 14 April 2011

J . -K. FU AND S.-M. LlANG

Figure 1 Configuration of a typical SOC projectile, Ref. I .

the shell; thus, the reduction in base drag becomes a major and direct consideration for improving the performance of a projectile. A significant amount of work in reducing the base drag for axisymmetric bodies has been done in the past three decades mainly by means of b o a t t a i ~ i n ~ ~ - ~ and base bleed4-7. Although in the transonic regime an additional wave drag is induced by shock wave developed on the boattail portion, nevertheless, boattailing can effectively reduce the base drag, leading to a total drag reduction, due to the fact that the area of the base region and extent of flow expansion at the base corner are reduced5-'. While, the base bleed method is by injecting a small amount of mass flow into the base region in order t o increase the base pressure, and leads to a base drag reduction. This concept of mass injection a t the projectile base has been widely studied for supersonic

but fewer data are available in the transonic flow r e g i n ~ e ~ - ~ . Thus, the accurate prediction of the effect of base bleed, in terms of bleed quantity and bleed area, on the total aerodynamic drag becomes essential for the transonic projectile shell design.

In the past fifteen years, earlier experimental research programs were predominantly in the supersonic regime; but in recent years1-*, efforts have been extended to the transonic regime. However, in experiments, the individual drag components, especially the base drag and skin friction drag, are difficult t o measure, and the cost of experimental measurements (together with limited transonic fa'dities) are very expansive. Due to the recent advances in computer processors, significant progress in numerical procedures3s4"3~14 has been achieved; thus, numerical simulation is a 'good' approach for predicting all individual drag components for projectiles on a cost-effective basis.

In the transonic regime, the flow over a SOC projectile is quite complicated


DRAG FOR A TRANSONIC PROJECTILE 25 1

mainly due to the instantaneous existences of the fast flow expansion at nose- cylinder juncture, the interaction between shock waves and turbulent boundary layer, and the flow separation near the base corner followed by a recirculatory flow in the base region. Since this complex flow structure often causes numerical instability due to improper grid distribution and/or numerical scheme, such as the Beam-Warming schemels which produces spurious oscillation near the shock waves. Thus, it is needed to adopt a robust scheme and proper grids for obtaining an accurate calculation of transonic flows over the projectiles.

In addition to producing the oscillating solution near the shock wave, the traditional central-differencing schemes generally often require special tuning of the artificial viscosity to reduce the numerical oscillations. In order to overcome these difficulties, an implicit, diagonalized, symmetric Total Variation Diminishing (TVD) ~ c h e r n e ' ~ . ~ ' , accompanied by a suitable grid1', is employed to solve the thin- layer axisymmetric Navier-Stokes equations1' coupled with the Baldwin-Lomax turbulence modelz0. The linearized conservative implicit (LCI) form of TVD schemez1 is adopted in this study. The TVD scheme not only can avoid numerical oscillations but also can satisfy the entropy condition so that no expansion shock is capturedzz."'. Like many shock capturing schemes, the TVD scheme has a second-order accuracy in regions of smoothness and is first-order accurate a t points of extrema. The assessment of TVD schemes for inviscid- and turbulent-flow computations by Chen et al.% revealed that the symmetric TVD method requires less CPU time per iteration and has a faster convergence rate than the upwind TVD schemes in two-dimensional computations, although both symmetric and upwind TVD schemes have almost identical accuracy and stability limit.

The objectives of this research are (1) to investigate the flow structure (especially in the base region) of the SOC projectile with and without base bleed in transonic regime at zero angle of attack, and (2) to study systematically the effects of bleed quantity (I) and bleed area ratio (G) on the variations of nose (or head) pressure drag (CDH), viscous drag (CDV), base drag (CDB), respectively, and finally the total drag (Cm) coefficients. But, the combined effect of boattailing and base bleed is not considered here.

2. MATHEMATICAL FORMULATION

2.1. The Axisymmetric Thin-Layer Navier-Stokes Equations

For an axisymmetric projectile at zero angle of attack, the equations governing the flow are the azimuthal-invariant Navier-Stokes equations, which can be obtained from the three-dimensional Navier-Stokes equations. Figure 2 shows the notation for the Cartesian coordinates x, y , z, the cylindrical coordinates x, 6 , R and the transformed coordinates t , q , 5.. where E , q and r represent the longitudinal, the circumferential and the near normal coordinates, respectively. For high Reynolds number flow, the transformed nondimensional axisymmetric thin-layer Navier-


252 J . -K. FU AND S.-M. LlANG

(a) Cone-cylinder body (b) x = const. plane

(c) 4 = const. plane Figure 2 Axisymmetric body and coordinate system.

Stokes equations can be written as19

where

with


DRAG FOR A TRANSONIC PROJECTILE

and

Here U, V, Ware the contravariant velocity components, and the Jacobian, J, is defined as

Let computations be performed on the 4 = 0 plane. Then, in equation (2d), R = z , Rt = zr and Rr = zr for 4 = 0. The metric terms &, qx, qr and Si. are zero when equations (2a) through (2c) are used, and the rest of the metric terms can be found in Reference 19.

Although equation (1) contains only two spatial derivatives, it retains all three momentum equations. In particular, the circumferential velocity is not assumed to be zero, and computations for spinning projectiles or swirling flow can be accomplished.

It should be noted that the thin-layer approximation has been widely applied for computing high Reynolds number turbulent flows over proje~tiles4.L23'4325 . The validity of the thin-layer approximation was studied by Degani and schiffZ5 who found that less computing time (about 11%) is required for the thin-layer approximation compared with the full Navier-Stokes equations, while the change (or improvement) in the solution (including the separated region) of the full Navier-Stokes equation is very small (less than 0.5%). The two-layer algebraic turbulence model of ~a ldwin -~omax~O is employed. The advantage of this turbulence model is that it is simple to implement and there is no special need to determine the outer edge of the boundary layer.

2.2. Grid Generation

In general, it is convenient to use a 0-type grid for the projectile with a basez6. In this study, the grid generation procedure is based on the numerical approach of Sorenson's elliptic solverz7 and a simple tangent stretching interpolation method2'. Once the coordinates of the body and the outer boundary are specified, the elliptic solver is used to generate a coarse grid (in the r direction). Then, the stretching interpolation is used to obtain a final grid, as shown in Figure 3. The grid for the whole computational domain and the local grid near the base region are shown in Figures 4 and 5, respectively. Grid points are clustered near the body surface as well as regions with high gradients, and stretched in the far field.



Figure 3 Local grid near a SOC projectile.

Figure 4 Overview of a 110 x 60 0-type projectile grid.



Figure 5 Local grid near the base corner.

3. NUMERICAL PROCEDURE

3. I . Approximate Factorization

Approximate factorization method of Beam and warmingI5 is applied to equation (I) , to obtain a semi-finite-difference equation

where h = A t , and A, e and M are the Jacobian matrices defined as

Equation (3) contains two implicit operators in which each operator will produce a block tridiagonal matrix. The inversion of the block-tridiagonal matrices is the most time-consuming part of the computation. The overall computational work of this part can be reduced by introducing a diagonalization of the blocks in the implicit operators29 and by neglecting the viscous flux ~ a c o b i a n ~ ' ~ " . Since A and e can be diagonalized, A = T€A€F' and e= T ~ A ~ T F ' , where matrices Tc and Tr consist of the right eigenvectors of A and e, respectively, and the elements of diagonal matrices A€, A' are the corresponding eigenvalues. After neglecting the


256 J . -K. FU AND S.-M. LlANG

viscous Jacobian on the implicit side and factoring the similarity matrices TE and Tr outside the spatial derivatives ac and a,, from equation (3) one obtains

3.2. The Symmetric TVD Method

Let t = n At, E = J Af, { = /A{, A'= AtlAf, Xr = At/A{, and ajm+1/2, Tj+1/2, _ T ~ I / Z

denote the quantities a;", Ti, Tcl evaluated at &j+l/z,i. Hence, the values of Qj+l/2,r and & . r + m are evaluated at the cell surfaces, respectively, using Roe's averaging procedure32. The final diagonalized form of the symmetric TVD scheme can be written in the

The detailed form of M can be found in Reference 33. The term R^(&" ) on the right hand side of equation (5a) is of the form

where the numerical flux function Ej+ 1/2.1 can be expressed asz1

and Dj+1/2 is the explicit dissipation term. The definition of D is given in Reference 16.

4. BOUNDARY AND INITIAL CONDITIONS

4. I. Wall Boundary Conditions

The body surface is mapped t o the constant { line ({ = 0) in the computational domain. On the impermeable surface of a rigid body without spinning, the no slip condition must be satisfied. While for projectiles with base bleed, the subsonic mass injection boundary condition is used34. It should be noted that since the amount of the bleeding air flow is small such that the subsonic mass injection boundary condition corresponds (or reduced) to a linear extrapolation if the change in pressure (Ap =p, - P I ) between the wall pressure, p, and the pressure a t first grid off the wall, pl, is small (about 10% of p l ) . Furthermore, the bleeding air flow used


DRAG FOR A TRANSONIC PROJECTILE 257

in the present study is assumed t o have a temperature close to that in the base region such that the effect of temperature of the bleeding air flow is negligible.

Due to the absence of a measured wall bleed distribution, a uniform distribution of surface bleed is employed. The amount of air flow injected into the base region is specified on the body surface by the mass flow rate mj (=pjujAj), where A j denotes the bleed area. Rather than specifying mj, however, it is customary and convenient to specify the bleed quantity, I, which is defined as4

where 3 is the area ratio Aj/Ab, with Ab the base area. Note that both the bleed quantity I and the area ratio a are two parameters in the base bleed method for the base drag reduction. Given I, 3, M, and the extrapolated density p j from the first interior point off the body surface, the injection velocity uj can be calculated from equation (8), and v and w are set equal to zero.

The pressure on the body surface can be obtained from the normal momentum equation l9

where n denotes the normal direction of the body surface. The total energy e is obtained from

4.2. Far Field Boundary Condition

The far field boundary condition is imposed at !: = r,,, (line ED) for an 0-type grid system. On the inflow boundary (Wf,,. < 0), all physical variables, Qi, are fixed at freestream values for both supersonic and subsonic flows. On the outflow boundary (Wf,, > 0), all variables Qi are extrapolated from the first interior point off the outer boundary for M, > 1. While for M, < 1, the same conditions used in supersonic flow are used except that the total energy e is obtained from equation (10) with fixed freestream pressure p, and the other extrapolated quantities. Since the downstream boundary adopted in the present study is far enough (25 calibers) such that the effect due to the base bleed on the downstream boundary condition is small and can be neglected, the simple extrapolation for all of the variables is used on the downstream boundary (line CD) for both bleeding and non-bleeding cases.

4.3. Centerline Boundary Condition

The symmetric condition is applied on the centerlines o f projectile ahead of the nose ([= 0) and aft of the base ((= (,,,). A second-order extrapolation is used


258 J . - K . FU AND S . - M . LlANG

such that alp = & ( p u ) = ac(pv) = are = 0, and the momentum in the z-direction (pw) is set equal to zero.

4.4. Initial Condition

In this study, all flow variables are set to the free-stream values initially, then the boundary conditions are enforced gradually during the iteration process. This slow impulsive start accompanied with a smaller value of the Courant number, CFL, helps to reduce the initial transient errors, especially for viscous flows, and to ensure that no numerical instability is produced during the initial computing process.

4.5. Selection of Time Step

To maintain numerical stability, the required time step generally is a function of local grid spacings. Thus, if a constant time step is used for solving the transonic flows on a highly stretched grid, the smallest time step associated with the smallest spacing must be used, and the CPU time required for the computations will be relatively high. For obtaining a fast converged steady-state solution, a spatially varying time step is used such that more uniform Courant numbers are used throughout the flow field. This is an effective strategy for a stretched grid system. The stability criterion for an explicit scheme is used with a larger Courant number, CFL. The time step varies according to3'

where a is the local sound speed. In the first 50 iterations, a smaller CFL (around 1 .O) is used for avoiding the occurrence of numerical instability. As the iteration proceeds, the Courant number is increased to a larger value (between 1.5 and 3.0) for fast convergence and less CPU time.

5. RESULTS AND DISCUSSION

Transonic turbulent flows over a zero angle-of-attack SOC projectile with base bleed are considered. The projectile model has a 3-caliber secant-ogive-nose followed by a 3-caliber cylinder. All the present computations were performed on an ALLIANT computer FX2800. A series of computations were based on the conditions of a stagnation temperature of 580' R and a Reynolds number of 4.6 x lo6 based on the model, length, L. The axisymmetric projectile flows are computed on the qi = 0 plane, and the flow domain chosen has an outer boundary about 25 calibers from the projectile. The computational grid used, as shown in Figure 4, has 110 points in the streamwise direction and 60 points in the direction normal to the projectile surface. This grid system is clustered near the body surface with a minimum spacing of A r,i, = 4.2 x lO-'d to ensure that there are at least two



grid points inside the laminar sublayer. Steady-state solutions are achieved when the residual measured by the root-mean-square error in all five conservative variables is less than 1.5 x The transonic flows over the same SOC projectile for the case of no base bleed have been studied by the authors36 using the present TVD scheme, and obtained reasonable and accurate results for all the drag components and total drag predictions. These computed results for the case of no base bleed are used for the comparison with the bleed case in this study. Six Mach numbers ranging from 0.9 to 1.2 are chosen here, and the base bleed takes place over 90% of the base area.

5.1. Effect of Bleed Quantity

For determining the effect of mass flow rate of base bleed on the base drag reduction, five different bleed quantities ( I = 0.025, 0.05, 0.10, 0.15 and 0.20) with a bleed area ratio of 0.9 are chosen, where I is defined in equation (8). Figures 6 and 7 show the Mach contours and pressure contours of a SOC projectile at Mach number of 0.96 with bleed quantities of 0 and 0.1, respectively. It is found that there exist two strong expansions at the nose-cylinder juncture and the base corner, respectively. For the both cases, the flow structure are almost identical except in the base region due to the base bleed. The predicted location of a shock wave on the cylinder is in close agreement with the shadowgraph reported by Kayser and whiton'. The supersonic region near the base corner (due to the fast flow expansion) is reduced by the base bleed, and the pressure (or pressure gradient) in the base region is increased (or decreased) as bleed quantity increases. The velocity vector fields in the base region for the corresponding cases are shown in Figure 8. It can be seen that the core of the recirculatory flow moves toward downstream for the bleed case.

The computed distributions of the pressure coefficient, C,, on the projectile surface for the above cases are shown in Figure 9. It is found that the pressure coefficient on the projectile surface is independent of the base bleed except on the base where the pressure is increased. The distribution of C, beyond the base (x = 6) represents the pressure coefficient along the symmetry axis, [ = Cmar The pressure along the symmetry axis is increased from the base, reaching a maximum at the reattachment point, and followed by a downstream recovery. The predicted reattachment points are at about x = 7.3 and 7.6 for I = 0 and 0.1, respectively. Moreover, the peak of the pressure distribution along the symmetry axis is decreased after base bleed. For the non-bleeding case (I= O), as shown in Figure 9(a), the Kayser's experiment2 gives higher value of C, on the base surface, this may be due to a sting is mounted on the projectile base and hence cause some disturbances on the base pressure. Figure 10 represents the comparison of the base drag coefficient, C m , for I= 0 case. The present computed results at all chosen Mach numbers were well predicted compared with those by the MCDRAG code". Some large discrepancies between the Kayser's experiments2 and the predictions by the Beam-Warming schemeL3 at lower Mach number were observed. Note that, in Kayser's experiments, the higher pressure on base surface due to the existence of


(b) Pressure contour

Figure 6 Transonic flow over a projectile, M, = 0.96, I = 0.



(a) Match contour

Figure 7 Transonic flow over a projectile with base bleed, M, = 0.96, G = 0.9, I = 0.1.


1. -K. FU AND S.-M. LIANG

( a ) I = 0

0 I 5.5 6.0 6.5 7.0 7.5

X (b) I = 0 . 1

Figure 8 Velocity vector field in the base region, M, = 0.96, a = 0.9.



0.6 8.0

0.6 8.0

. 0.4-

0.4-

Present result - 6.0

0.2 -

0.0-

Cp on t h e base - 4.0e -0.2-

-0.4 - 2.0

-0.6-

-0.8 -0.6-0.4 -0.2 0.0

-0.6 CP 0.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0

X (a) I = 0

A E X P E R I M E N T ~ I = O ] Ref. 1

Present result

- 8.0 0.2 -

a u Cp on t h e base ; 4.0 #

-0.2

-0.4

-0.6 -0.8-0.6-0.4 -0.2 0.0

'

- - 0.8,< . . . . . . . . . . . . . . . . . . . . . . . 0.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0

X (b) I = 0 . 1

Figure 9 Comparison of pressure distribution on a projectile. M, = 0.96, G = 0.9 .

o E X P E R I M E N T [ I = ~ ] Ref. 1 '



0.35 -*- M C D R A C (371 Q.

- - - E X P E R I M E N T A L . [ ~ ~ +- - B E A M - W A R M I N G [13

0.30- Present result

MACH N U M B E R Figure 10 Comparison of base drag coefficient of a projectile, I= 0.

the sting leads to a reduction in base drag. And, Hsu et a1.I4 commented the underpredicted base drag (for M, < 0.98) obtained by the Beam-Warming scheme is due to an oscillation in the pressure distribution along the base even though a converged solution was obtained.

The computed head pressure drag coefficient, CDH and viscous drag coefficient, CDV are found to be independent of the base bleed for Mach numbers from 0.9 to 1.2 (see Table 1). The variations of the base drag coefficient (CDB) and the total

Table 1 Comparison of drag components and drag reduction with different base bleed rates, 8 = 0.9

Mach No. Bleed Rate Head Drag Visc. Drag Base Drag Total Drag CDB ACm ACDB I (Q70)

- - M, CDH CDV CDB Cm Cm Cool,-o CDB(I-o



0 . 0 5 i . . , . , , ~ , , l , , , , l , , . , l , , , , , , . , , I

-0.05 0.00 0.05 0.10 0.15 0.20 c BLEED RATE, I

Figure 11 Variation of base drag coefficient of a projectile, G = 0.9.

0 . 1 1 -0.05 0.00 0.05 0.10 0.15 0.10 C

BLEED RATE, I Figure 12 Variation of total drag coefficient of a projectile, 3 = 0.9.


J:K. FU AND S.-M. LlANG

. I . I , . . . . I . . . , , , , , . ( r ! . , , . . , . 0.8 0.9 1.0 1.1 1.2 1.3

MACH N U M B E R Figure 13 Variation of base drag coefficient, G = 0.9.

MACH NUMBER Figure 14 Variation of total drag coefficient, G = 0.9.



drag coefficient (Cm) with base bleed for different Mach numbers are shown in Figures 11 and 12, respectively. It can be seen that both the base drag and total drag decrease with increasing bleed quantity. The variations of the base drag coefficient and total drag coefficient with various freestream Mach numbers at different bleed quantities are shown in Figures 13 and 14, respectively. These results indicate that the base bleed used for reducing the base drag is more effective for the supersonic flows than that in the subsonic flows. As listed in Table I, for larger bleed quantity (say, I = 0.2), the base drag reduction is over 50% for all three Mach numbers, and the weights of the base drag reduce from 68.6%, 67.0% and 58.8% for the case of no base bleed to 51.4070, 50.4% and 38.8% of the total drag at Mach numbers 0.9, 0.96 and 1.2, respectively. While, the reduction in the total drag is about 35.7%, 34.4% and 32% for Mach numbers 0.9, 0.96 and 1.2, respectively. It should be noted that for the bleeding cases around I 2 0.1, there may be theoretically inconsistent for the application of the thin-layer concept to describe the base flow. However, for I = 0.1 case, it is found that the computed base drag coefficient from the full Navier-Stokes equations computation differs from the result of the thin- layer approximation only by I%, but the former needs larger computing time (about 15%).

5.2. Effect of Bleed Area Ratio

To study the effect of bleed area on the base drag reduction, three different bleed area ratios, B = 0.9, 0.6 and 0.3 were selected, where B is defined in Section 4.2. The calculations are performed for case of Mach number 0.96. The Mach contours and pressure contours of a SOC projectile with bleed quantity of 0.1 for bleed area ratio of 0.3 are shown in Figures 15(a) and 15(b), respectively. Comparing with the result of the B = 0.9 case as shown in Figure 6, the supersonic zone near the base corner is further reduced and the pressure in the base region is increased as bleed area ratio decreases. This is due to the higher injection speed resulted from a smaller bleed area for fixed bleed quantity, and hence a higher momentum is injected into the base region. The velocity vector field in the base region for the corresponding case shown in Figure 16, it is found that the recirculatory flow is disappeared. Figure 17 shows the corresponding computed distributions of the pressure coefficient on projectile surface. The result shows that the pressure distribution on the projectile surface is also independent of the bleed area ratio; but on the base the pressure distribution increases with decreasing bleed area ratio. The peak of the pressure distribution on the symmetry axis, [ = [,,, decreases with the bleed area ratio. The predicted head drag and viscous drag are almost independent of the bleed area ratio (see Table 2).

The variations of the base drag coefficient and total drag coefficient with base bleed for different bleed area ratios are shown in Figures 18 and 19, respectively. It can be seen that both the base drag and the total drag decrease with bleed area ratio and with increasing bleed quantity. The reduction rates in both the base drag and total drag increase with decreasing bleed area ratio for bleed quantity from 0 to 0.2. From Table 2, it is seen that the reduction in base drag, for I = 0.1, is 40%


268 J.-K. FU AND S.-M. LIANG

(:::-r--------:------------..,<TI

on

Cl

N

c::~+l...:....O...LiJI.O~~0=--[r-.0-'2.-0-J~:-0-1-:-~-0-5~~-0-6-1"'~0=>'-7r-,.0~8=. 0

X

(a) Match contour

c:)-r- -;--__~-----..,en

C::J

co

C:J

r.....

C:J

10

(:J

1'1

':.1.0 0.0 1.0 2.0 J:O i~O 5.0 6.0 7.0 8.0

X(b) Pressure contour

Figure 15 Transonic flow over a projectile with base bleed, Moo = 0.96, w= 0.3, 1=0.1.



---""'=- - - --=- --

------ - -= == .." --~--~::::.= -o --=- -----.- - -}-._--~-=---===.::::. -

----...... ~--~_---...--

~---::---

~~+-==== ~ =-== _:::::ss:-: __ -

co

269

6.0 8..5

x1.0 7.S 5.0

Figure 16 Velocity vector field in the base region, Moo = 0.96, 1= .1, w= 0.3.

(L8 ----- 8.0

0.4- EXP ER1MENTp=O.O;Present result 1=0.1

6.00.2

':0.0

a.4.0 ~U on the base

-0.20.50,4

-0.4- 0.3

0::: 0-.2 2.0

0.1-0.6

0.0-0'.8 -0.6 -0.4 -0.2 0.0---- Cp

-0.8 -- 0.0--r-T""""T~-r-r-

0.0 2.0 4.0 6.0 8.0 ·10.0 12.0

X

Figure 17 Comparison of pressure distribution on a projectile, Moo = 0.96, w= 0.3.


270 1 . - K . FU AND S.-M. LLANG

Table 2 Comparison of drag components and drag reduction with different base bleed rates, M, = 0.96

Bleed Area. Bleed Rate Head Drag Visc. Drag Base Drag Total Drag CDB ACDO ACDB t% I (%) CDH CDV CDB Cm C,(,.o -0

- - - - - - -

for Ci = 0.6 case and 64.2% for = 0.3 case. For the case of G = 0.3 and I = 0.2, the reduction in base drag can be as high as 88.3% and the base drag occupies only about 19% of the total drag instead of about 67% for no base bleed; the reduction in total drag is about 60%.

It should be noted that at fixed bleed quantity, a smaller bleed area ratio produces a same effect as a larger bleed quantity does. However, a large bleed quantity needs large pumping power; thus, in order to effectively reduce the base drag by the base bleed method, a smaller bleed area ratio (0.3 6 G 6 0.6) accompanied with a small bleed quantity ( I 6 0.1) is suggested. But, the bleed area ratio can not be unlimited small since it also needs tender of a higher pumping pressure for same bleed quantity.

. . . . I . " ' , , " . ' I ' - . . , . ' r ' , " . '

-0 .05 0 .00 0.05 0.10 0.15 0.20 I

BLEED RATE, I Figure 18 Variation o f base drag coefficient, M, = 0.96.


DRAG FOR A TRANSONIC PROJECTILE 27 1

0 . 0 5 -0 .05 0 .00 0 .05 0.10 0.15 0 . 2 5

BLEED RATE, I Figure 19 Variation of total drag coefficient, M, = 0.96.

6. CONCLUSIONS

The transonic flows over a secant-ogive-cylinder projectile with base bleed have been successfully simulated by solving the axisymmetric thin-layer Navier-Stokes equations, using an implicit, symmetric TVD scheme. The computed results show that the base drag and total drag can be reduced by increasing the bleed quantity or by decreasing the bleed area ratio at fixed bleed quantity (i.e. increasing the injection speed). For the case of a = 0.3 and I = 0.2, the reductions in base drag and total drag can be as high as about 88% and 60%, respectively. It is suggested that a smaller bleed area ratio (0.3 $ G $ 0.6) associated with a small bleed quantity (I < 0.1) can obtain a better effect in the aerodynamic drag reduction. It is concluded that the present TVD scheme is robust enough for the predictions of transonic flows over a projectile with or without base bleed, and can be an effective tool for the design of high-performance projectiles.

I . Kayser, L. D. and Whiton, F. (1982) "Surface pressure measurements on a boattailed projectile shape at transonic speeds", U.S. Army. Ballistic Research Laboratory Technical Rept. ARBRL-MR- 03161.

2. Kayser, L. D. (1984) "Base pressure measurements on a projectile shape at rnach numbers from 0.91 to 1.20", U.S. Army, Ballistic Research Laboratory. Aberdeen Proving Ground, Md., ARBRL-MR- 03353 (AD A141341).


272 J.-K. FU AND S.-M. LlANG

3. Sahu, J.. Nietubicz, C. J. and Steger, J. L. (1982) "Numerical computation of base flow for a projectile at transonic speeds", AIAA Paper No. 82-1358.

4. Sahu, J., Nietubicz, C. J. and Steger, J . L. (1985) "Navier-Stokes computations of projectile base flow with and without mass injection", AIAA Journal, 23 (9), 1348-1355.

5. Sedney, R. (1966) "Review of base drag", U S . Army Ballistic Research Laboratory, Aberdeen Proving Ground. Md., Rept. 1337 (AD 808767).

6. Tanner, M. (1975) "Reduction of base drag", Prog. Aerospace Sci., 16 (4). 369-384. 7. Sykes, D. M. (1970) "Cylindrical and boat-tailed afterbodies in transonic flow with gas ejection",

AIAA Journal, 8 (3), 588-590. 8. Schetz, J . A., Billig. F. S. and Favin, S. (1980) "Approximate analysis axisymmetric supersonic base

flows with injection". AIAA Journal. I8 (8). 867-868. 9. Addy, A. L. (1970) "Thrust-minus-drag optimization by base bleed and/or boattailing", Journal of

Spacecraft, 7 (11). 1360-1361. 10. Bowman, J. E. and Clayden, W. A. (1968) "Boat-tailed afterbodies at M = 2 with gas ejection",

AIAA Journal. 6 (lo), 2029-2030. 1 1 . Sullins, G. A,, Anderson. J. D. and Drummond, J . P. (1982) "Numerical investigation of

supersonic base flow with parallel injection", AlAA Paper 82-1001. 12. Deiwert, G. S. (1984) "Supersonic axisymmetric flow over boattails containing a centered propulsive

jet". AlAA Journal, 22 (lo), 1358-1365. 13. Sahu. J . (1986) "Drag predictions for projectiles at transonic and supersonic speeds". U S . Army

Ballistic Research Laboratory, Aberdeen, Proving Ground, Md., Memorandum Report BRL-MR- 3523 (AD A171462).

14. Hsu. C. C.. Shiau. N. H . and Reed, C. W. (1988) "Numerical simulation of transonic turbulent flow past a real projectile", AlAA Paper 88-0218.

15. Beam, R. M. and Warming, R. F. (1978) "An implicit factored scheme for the compressible Navier Stokes equations", AIAA Journal. 16 (4), 393-402.

16. Shiau, N. H. and Hsu, C. C. (1988) "A diagonalized TVD scheme for turbulent transonic projectile aerodynamics computation", AlAA Paper 88-0217.

17. Yee, H. C. (1985) "On symmetric and upwind TVD schemes", Proceedings of the 6th GAMM Conference on Numerical Methods in Fluid Mechanics. or NASA-TM 86842.

18. Fu, J. K. (1991) "A numerical study on drag reduction of turbulent transonic flow over a projectile", Ph.D. Dissertation, Institute of Aeronautics and Astronautics, National Cheng Kung University. Tainan, Taiwan. R.O.C.

19. Nietubicz. C. J.. Pulliam. T. H. and Steger, J . L. (1979) "Numerical solution of the azimuthal invariant thin-layer Navier-Stokes equations", AlAA Paper 79-0010.

20. Baldwin, B. S. and Lomax, H. (1978) "Thin-layer approximation and algebraic model for separated turbulent flows", AlAA Paper 78-257.

21. Yee. H. C. and Harten, A. (1987) "Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates", AIAA Journal, 25, 266-274. Also (1985) AlAA Paper No. 85-1513.

22. Harten, A. (1983) "A high resolution scheme for the computation of weak solutions of hyperbolic conservation laws", Journal of Compulalional Physics, 49, 357-393.

23. Yee. H. C.. Warming, R. F. and Harten. A. (1983) "Implicit total variation diminishing (TVD) schemes for steady state calculations", AlAA Paper No. 83-1902.

24. Chen, M., Hsu, C . and Shyy, W. (1989) "Assessment of TVD schemes for inviscid and turbulent flow", AlAA Paper No. 89-1795.

25. Degani, D. and Schiff. L. B. (1986) "Computation of turbulent supersonic flows around pointed bodies having crossflow separation". Journal of Compulafional Physics, 66, 173-196.

26. Steger, J . L., Nietubicz, C. J . and Heavey, K . R. (1981) "A general curvilinear grid generation program for projectile configurations". U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, Md., Memorandum Report ARBRL-MR-03142 (AD A107334).

27. Sorenson, R. L. (1980) "A computer program to generate two-dimensional grids about airfoils and other shapes by the use of Poisson's equation", NASA TM-81198.

28. Thompson, J. F. (1987) "A general three-dimensional elliptic grid generation system on a composite block structure". Computer Methods in Applied Mechanics and Engineering, 64, 377-41 1.

29. Pulliam. T. H. and Chaussee, D. S. (1981) "A diagonal form of an implicit approximate factorization algorithm", Journal of Compulolionol Physics, 39. 347-363.

30. Pulliam, T . H. (1984) "Euler and thin-layer Navier-Stokes codes: ARCZD, ARC3DU, Notes for CFD User's Workshop, Tullahoma, Tennessee.

31. Pulliam. T. H. and Steger, J . L. (1985) "Recent improvements in efficiency, and convergence for implicit approximate factorization algorithms", AlAA Paper 85-0360.



32. Roe, P. L. (1981) "Approximate Riemann solver, parameter vectors, and difference schemes", Journal of Computational Physics, 43, 357-372.

33. Hu, C. L. and Liang. S. M. (1989) "Numerical investigation of thrust-reversing nozzle using an implicit TVD scheme", AlAA Paper 89-2899.

34. Wang. J. C. T. (1993) "Design and development of airframe and propulsion system", Notes for Workshop on Institute of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan.

35. Anderson, W. K., Thomas, J. L. and van Leer, B. (1985) "A comparison of finite volume flux vector splittings for the Euler equations", AlAA Paper 85-0122(1985).

36. Fu. J . K. and Liang, S . M. (1992) "Computations of turbulent transonic flow over a projectile with or without spinning', Proceedings of the 34th Conference of the Aeronautical and Astronautical Society of Republic of China, Taiwan, Tainan.

37. McCoy, R. L. (1981) 'McDrag - A computer program for estimating the drag coefficients of projectiles", U S . Army, Ballistic Research Laboratory, Aberdeen Proving Ground. Md., ARBRL- TR-02293 (AD A0981 10).


numerical study of base bleed effects on aerodynamic drag for a transonic projectile

Documents