duct design in subsonic and supersonic

Transaction B: Mechanical EngineeringVol. 17, No. 3, pp. 179{193c Sharif University of Technology, June 2010

Duct Design in Subsonic and SupersonicFlow Regimes with and without Normal

Shock Waves Using Flexible String Algorithm

M. Nili-Ahmadabadi1;�, M. Durali2, A. Hajilouy-Benisi1 and F. Ghadak3

Abstract. In this investigation, the Flexible String Algorithm (FSA) used before for the inverse designof 2D subsonic ducts is developed and applied for the inverse design of subsonic and supersonic ducts withand without normal shock waves. In this method, the duct wall shape is changed under a novel algorithmbased on the deformation of a virtual exible string in a ow. Deformation of the string due to the local ow conditions resulting from changes in the wall geometry is performed until the target shape satisfyingthe prescribed walls pressure distribution is achieved. The ow �eld at each shape modi�cation step isanalyzed using an Euler equation solution by the AUSM method. Some validation test cases and designexamples in subsonic and supersonic regimes are presented here, which show the robustness and exibilityof the method in handling the complex geometries in various ow regimes. In the case of unsymmetricalducts with two unknown walls, the FSA is modi�ed to increase the convergence rate signi�cantly. Also, thee�ect of duct inlet and outlet boundary conditions on the convergence of the FSA is investigated. The FSAis a physical and quick converging approach and can e�ciently utilize ow analysis codes as a black box.

Keywords: Duct design; Subsonic; Supersonic; Normal shock; FSA; Internal ow; Inviscid.

INTRODUCTION

Duct Design, such as intakes, manifolds, duct reducers,compressor and turbine blades etc. is based onwall shape determination, so that the ow is opti-mum. Often, both Computational Fluid Dynamics(CFD) and design algorithms are involved in solvingan optimal shape design problem. The limitationsand computational cost of the design techniques arechallenging problems for present time computationaltechnology.

One of the optimal shape design methods is theSurface Shape Design (SSD). The SSD in uid owproblems usually involves �nding a shape associatedwith a prescribed distribution of surface pressure or

1. School of Mechanical Engineering, Center of Excellence inEnergy Conversion, Sharif University of Technology, Tehran,P.O. Box 14588-89694, Iran.

2. School of Mechanical Engineering, Center of Excellence inDesign, Robotics and Automation, Sharif University of Tech-nology, Tehran, 14588-89694, Iran.

3. Qadr Aerodynamic Research Center, Imam Hossein Univer-sity, Tehran, Iran.

*. Corresponding author. E-mail: [email protected]

Received 14 March 2009; received in revised form 20 December2009; accepted 22 February 2010

velocity. It should be noted that the solution of aSSD problem is not generally an optimum solution ina mathematical sense. It just means that the solutionsatis�es a Target Pressure Distribution (TPD) whichresembles a nearly optimum performance [1].

There are, basically, two di�erent algorithms forsolving SSD problems: decoupled (iterative) and cou-pled (direct or non-iterative). In the coupled solutionapproach, an alternative formulation of the problemis used, in which the surface coordinates appear (ex-plicitly or implicitly) as dependent variables. In otherwords, coupled methods tend to �nd the unknownpart of the boundary and the ow �eld unknownssimultaneously in a (theoretically) single-pass or one-shot approach [1].

The traditional fully coupled approaches trans-form the ow equations to a computational domainin which the unknown coordinates appear as depen-dent variables. Stanitz [2-4] solved two- and three-dimensional potential ow duct design problems usingstream and potential functions as independent vari-ables. Zanetti [5] considered two-dimensional and axi-symmetric Euler equations and mapped the physicaldomain to a �xed computational region. A novel directshape design method was proposed by Ashra�zadeh

180 M. Nili-Ahmadabadi et al.

et al. [6]. They basically showed that a fully coupledformulation of the SSD problem could be solved in thephysical domain using a simple extension of commonlyused CFD algorithms. Since the proposed direct designmethod does not need any transformation to or froma computational domain, it is applicable, in principle,to any ow model in 2 or 3D domains. Ghadak [1]extended the application of this method to the designof ducts carrying ows governed by non-linear coupledEuler equations.

The iterative (decoupled) shape design approachrelies on repeated shape modi�cations such that eachiteration consists of a ow solution followed by ageometry updating scheme. In other words, a seriesof sequential problems is solved, in which the surfaceshape is altered between iterations, so that the desiredTPD is �nally achieved [1].

Iterative methods, such as optimization tech-niques, have been by far the most widely used tosolve practical SSD problems. The traditional iterativemethods used for SSD problems are often based on trialand error or optimization algorithms. The trial anderror process is very time-consuming and computation-ally expensive and, hence, needs designer experience toreach minimum costs. Optimization methods [7,8] arecommonly used to automate the geometry modi�cationin each iteration cycle. In such methods, an objec-tive function (e.g., the di�erence between a currentsurface pressure and the target surface pressure [9])is minimized, subjected to the ow constraints whichhave to be satis�ed. Although the iterative methodsare general and powerful, they are often excessivelycomputationally costly and mathematically complex.These methods can utilize analysis methods for the ow�eld solution as a black box.

Other methods presented so far use physicalalgorithms instead of mathematical algorithms to au-tomate the geometry modi�cation in each iterationcycle. These methods are easier and quicker than theother iterative methods [1]. One of these physicalalgorithms is governed by a transpiration model inwhich one can assume that the wall is porous and,hence, the mass can be �ctitiously injected through thewall in such a way that the new wall satis�es the slipboundary condition. Aiming at the removal of non-zeronormal velocity on the boundary, a geometry updatedetermined by applying either a transpiration modelbased on mass ux conservation [10-15] or a streamlinemodel based on alignment with the streamlines [16]must be adopted.

An alternative algorithm is based on the residual-correction approach. In this method, the key problemis to relate the calculated di�erences between the actualpressure distribution on the current estimate of thegeometry and the target pressure distribution (theresidual) to required changes in the geometry. The art

in developing a residual-correction method is to �ndan optimum state between the computational e�ort(for determining the required geometry correction) andthe number of iterations needed to obtain a convergedsolution. This geometry correction may be estimatedby means of a simple correction rule, making useof relations between geometry changes and pressuredi�erences known from linearized ow theory. Theresidual-correction decoupled solution methods try toutilize the analysis methods as a black box [1].

Barger and Brooks [17] presented a streamlinecurvature method in which they considered the pos-sibility of relating a local change in surface curva-ture to a change in local velocity. Since then, alarge number of methods have been developed fol-lowing that concept. Subsequent re�nements andmodi�cations made the concept applicable to de-sign problems based on the full potential equa-tion [18], Euler equations [19] and Navier-Stokes equa-tions [20].

The main idea behind decoupling ow and ge-ometry solutions in inverse design, in most cases, isto take maximum advantages of the available analysismethods. Another advantage of decoupled solutionmethods is the fact that, in general, the constraintscan be implemented much more easily in a separategeometry update procedure than in a complete systemof equations for ow as well as geometry variables.

In this research, the Flexible String Algorithm(FSA) accomplished by Nili et al. [21] for inverse designof 2D subsonic ducts is supplemented and developed forsubsonic and supersonic ow regimes with and withoutnormal shock wave. In the case of unsymmetrical 2Dducts with two unknown walls, the FSA is modi�edto increase the convergence rate signi�cantly. Also,the e�ect of boundary conditions at the duct inlet andoutlet on the convergence of FSA is investigated.

The new feature of the FSA consists of consid-ering the duct wall as a exible string having mass.The di�erence between TPD and CPD at each shapemodi�cation step is applied to the string as an actual(external) force that accelerates and moves the string.Local acceleration of the string causes it to deformfrequently. Having achieved target shape, the di�er-ence between TPD and CPD vanishes and, �nally, thestring deformation is stopped automatically. Solvingthe string kinematic equations together with the owequations, at each modi�cation step, updates the ductshape so as to achieve the TPD.

In contrast to the other residual correction meth-ods using ow equations for inverse design problems,the FSA turns the inverse design problem into a uid-solid interaction scheme that uses the pressure conceptto deform the exible wall. Thus, it is more physicalthan the other methods. Also, the FSA convergesquickly and can easily incorporate an analysis code as a

FSA Inverse Design Method for Inviscid Internal Flows 181

black box. Therefore, higher computational e�ciencyand time saving will be expected.

FUNDAMENTALS OF THE METHOD

A 2-D ow �eld is assumed, as shown in Figure 1. Ifa exible string is �xed at point A in the ow, thepressure applied to the sides of the string deforms itto lay on a streamline passed through point A (A-B0curve). This phenomenon occurs because it is assumedthat no mass ux can pass across the string as astream line. For duct inverse design, the duct wallis considered as a exible string whose outer surfaceis exposed to the TPD and whose wetted surface isexposed to pressure resulting from passing ow throughthe duct.

Throughout the modi�cation procedure to achievethe target wall geometry, the unknown duct wall, like a exible string, is assumed to have a �xed starting pointand a free end point. The wall geometry is modi�edby the pressure di�erence between TPD and CPD.When the target wall shape is obtained, this pressuredi�erence logically vanishes.

In an asymmetric duct with two unknown walls,both upper and lower duct walls are modeled astwo strings deforming from an initial guess to thetarget shape. Considering a virtual string on theduct centerline and applying the di�erence betweenupper and lower pressure to the centerline string, theconvergence rate of the design algorithm increasessigni�cantly. At each modi�cation step, the ow �eldis analyzed using an Euler equation solution by theAUSM method [22].

MATHEMATICAL APPROACH

Governing Equations and BoundaryConditions for String

To derive the string kinematic relations, the string isapproximated by a chain with \n" links of equal lengthwith joints bearing no moment. Supposing uniformmass distribution along each link, the mass center islocated at its mid point. A free body diagram of

Figure 1. The string deformation in a 2D ow.

Figure 2. Free body diagram of an arbitrary link \i" ofthe chain.

an arbitrary link of the chain is shown in Figure 2.Assuming a 2-D motion of the chain for each link, threekinematic relations can be derived as follows:

1. Moment equation about the mass center of anarbitrary link:

12

(F j2xi + F j1xi )�si sin �i

� 12

(F j2yi + F j1yi )�si cos �i

=112�i(�si)3�i: (1)

2. Newton's second law in the x direction:

F j1xi � F j2xi ��p�iw�si sin �i = �i�siaxi: (2)

3. Newton's second law in the y direction:

F j1yi � F j2yi + �p�iw�si cos �i = �i�siayi: (3)

Relations between the linear accelerations of the �rstjoint of each link and its mass center are as follows:

�i�siaxi = �i�si�aj1xi � 1

2�i�si sin �i

�12!2i �si cos �i

�; (4)

�i�siayi = �i�si�aj1yi +

12�i�si cos �i

�12!2i�si sin �i

�: (5)

Equations 6 and 7 indicate the relations between thelinear accelerations of the two consecutive joints of eachlink.

aj2xi = aj1xi � �i�si sin �i � !2i �si cos �i; (6)

aj2yi = aj1yi + �i�si cos �i � !2i �si sin �i: (7)


Furthermore, the consistency equations at each jointare as follows:

aj2xi = aj1xi+1; aj2yi = aj1yi+1;

F j2xi = �F j1xi+1; F j2yi = �F j1yi+1: (8)

The boundary conditions of the chain (string) include�xed starting and free end points. Therefore, zeroacceleration and force are attributed to the startingand end points, respectively.

aj1x1 = aj1y1 = 0; (9)

F j2xn = F j2yn = 0: (10)

To study the chain kinematics, it is enough to calculatethe angular acceleration of each link (�i) and there is noneed to calculate the other unknowns such as the forcesand linear accelerations components. After eliminatingthe forces and linear accelerations and solving thelinear system of equations, the angular accelerationsare calculated exactly. Then, the angular velocity (!i)and the angle change of each link (��i) are obtainedas follows:

!t+�ti = !ti + �i�t; (11)

��i = 1=2�i�t2 + !i�t: (12)

Starting from the �rst link toward the end one, the newpositions of joints (j + 1) are obtained by adding theangle change of each link, with respect to the calculatedposition of the previous joint (j).

The solution starts with an initial guess such thatthe duct's main characteristics (e.g., length and inletor outlet area) are known and �xed. In this method,one of the joints coordinates (say x) is �xed.

xt+�tj = xtj ; (13)

yt+�tj+1 = yt+�t

j + �si sin(�i + ��i): (14)

In the case of a high curved duct such as an elbow,the mean length of the duct is known. Therefore,Equation 15 is used instead of Equation 13.

xt+�ti+1 = xt+�t

i + �si cos(�i + ��i): (15)

Applying Pressure Di�erence to the String

In order to converge the FSA design procedure, it isvery important as to how the di�erence between TPDand CPD is applied to the string.

In supersonic ow regimes, the ow informationjust transfers to the downstream. Keeping this physicalphenomenon in mind, the TPD and CPD applying

to the center of each link have to be shifted to thedownstream joint (Equation 16). Therefore, the inletpressure is shifted to the �rst point of the wall, andthe back pressure (pback) is eliminated from the wallpressure distribution. This procedure for subsonic owregimes is exactly vice versa (Equation 17). In otherwords, back pressure is shifted to the end point of thewall and the inlet pressure is eliminated from the wallpressure distribution.

pj+1 = pi; i = j = 0; n; (16)

pj = pi; i = j = 1; n+ 1;

i : index of each link; j : index of each joint: (17)

In subsonic ow regimes, the boundary conditionsare duct inlet Mach number and outlet pressure.Therefore, at each shape modi�cation step, the outletpressure remains constant, while inlet pressure changesaccording to the pressure in the �rst interior cell. As werequire the string starting point to remain stationary,�p�rst-joint must be zero. Thus, the pressure di�erenceapplied to the string at any other point must be gaugedwith respect to the pressure di�erence at the �rst joint.This is shown in Equation 18.

In supersonic ow regimes, pressure and Machnumber are �xed as inlet boundary conditions, andoutlet pressure is calculated according to the endinterior cell. In other words, during the shape mod-i�cation, inlet pressure remains constant; equal topin. Therefore, there is no need to gauge pressure(Equaiton 19).

�pj = (pTarget(j) � pTarget(1))� (pj � p1);

:: j = 1; n+ 1 :: pn+1 = pTarget(n+1) = Pback; (18)

�pj = (pj � pTarget(j));

:: j = 1; n+ 1 :: p1 = pTarget(1) = Pinlet: (19)

The main idea of the method is that for internalsupersonic ow regimes, the TPD and CPD are ap-plied along the outer and inner surface, respectively,while, for internal subsonic ow regimes, it is viceversa.

In ducts with both subsonic and supersonic owregimes, such as a supersonic nozzle with normalshock or convergent-divergent nozzles where the owis subsonic, the pressure di�erence is calculated fromEquation 18 and gauged with respect to the �rst jointwhere the ow is supersonic; it is calculated fromEquation 19. Finally, the pressure di�erence appliedto each link of the string is obtained from the following


equation:

�p�i = (�pj + �pj+1)=2; :: i = j = 1; n;

i : index of each link; j : index of each joint: (20)

FSA Design Procedure

Figure 3 shows how the string equations are typicallyincorporated into existing ow solution procedures.The computed pressure surfaces are normally obtainedfrom partially converged numerical solutions of the owequations. During the iterative design procedure, asthe CPD approaches the TPD, the force applied to thestring gradually vanishes and, at the �nal steps, thesubsequent solutions of the string equations yield nochanges in the duct surface coordinates.

VALIDATION

For validation of the proposed method, a given con-�guration, such as a supersonic nozzle, convergent-divergent nozzle or a 90-deg bended duct, is analyzedto obtain the solid wall pressure distribution. Then,these pressure distributions are considered as our TPDfor the SSD problems.

In all test cases studied here, the iterationswere stopped after the residuals were reduced by 3orders of magnitude in which residuals are de�nedas:

P j�pj=�pI.G.j j. After each geometry modi�cation

step, the analysis code is run until the residuals werereduced by 1 order of magnitude. The residual of theanalysis code is de�ned as the normalized changes inconserved ow variables in which the normalizationis performed by the residual of the �rst iteration(�j(Qn+1 �Qn)=QI.G.j).

Reducing the residuals 3 orders of magnitude forthe design algorithm and 1 order of magnitude for theanalysis code is enough to con�rm the required conver-gence so that the di�erence between calculated and tar-get shapes cannot be recognized. The design algorithmand analysis code show their capabilities to reduce theresiduals up to 6 orders of magnitude. But, during de-sign problems, any excessive decrement of the residualsjust increases the computational cost and time.

Here, the robustness of the design method isconsidered as the ability to use any initial guessesin each arbitrary computational grid. An idealrobust method should work well, regardless of theresolution of the computational grid and the initialguess.

Wind Tunnel Supersonic Nozzle

The �rst case is an ideal supersonic nozzle (Figure 4)which is extensively used in supersonic wind tunnels. Inorder to verify the capability of the proposed method,the ow �eld of the supersonic nozzle is analyzedand the wall pressure distribution is determined. Themethod should converge to this shape from an initial ar-bitrary shape if the goal is set to be the calculated (de-sired) pressure distribution. Initiating from a straightduct with a constant area section, the design programalgorithm is converged only after 60 modi�cation steps.The wall pressure distributions of the initial and �nalshape are shown in Figure 5. Figure 6 shows themodi�cation procedure from initial guess to the targetshape, in which the shape modi�cation procedure isaccomplished from upstream to downstream. The inletMach number and pressure have been set to 1.01 and1 bar, respectively. Overall, a computational grid of

Figure 3. Implementation of the inverse design algorithm.


Figure 4. Supersonic nozzle geometry with its grid.

Figure 5. Wall shape modi�cation procedure from aparallel duct to the ideal supersonic nozzle.

Figure 6. Wall pressure distribution of initial guess andtarget shape of the supersonic nozzle.

40 � 16 is used for the analysis code, as shown inFigure 4.

E�ect of Initial Guess

One of the outstanding capabilities of inverse designmethods is their full independency from the initialguess. Keeping this reason in mind, a convergent ductwith straight walls is considered as the initial guess fora supersonic nozzle. As shown in Figure 7, in contrastwith the presence of a high di�erence between initialguess and target shape, the initial guess is converged tothe target shape only after 90 iterations. An interestingpoint which shows the capability of the method is thatthe ow regime at primitive modi�cation steps includesboth subsonic and supersonic regimes. In Figure 8, the

Figure 7. Wall shape modi�cation procedure from aconvergent duct to the ideal supersonic nozzle.

Figure 8. Variation of wall Mach number distributionfrom a convergent duct to the ideal supersonic nozzle.


variations of wall Mach number from initial guess tothe target shape are shown.

Supersonic Nozzle with Normal Shock

The target pressure distribution for the second valida-tion test case is obtained from the solution of the owthrough the supersonic nozzle with normal shock withthe following characteristics: Entrance Area = 0.5 m2,Exit Area = 0.93 m2, Length = 4.6 m, Entrance MachNumber = 1.01, Entrance Pressure = 1 bar and BackPressure = 1.2 bar. Because a normal shock occursthrough the duct, a grid of 50 � 20 which is re�nedbeside the normal shock is used, as shown in Figure 9.Figure 9 illustrates the initial guess and the evolutionof the duct shape with their Mach number contour.Also, wall pressure distributions of the initial guess andtarget shape are shown in Figure 10. Although theinverse design of a duct with normal shock may haveless application, it well presents the great capability ofour design method. In Figures 11 and 12, a constantarea section duct and a convergent duct are used as theinitial guess for a supersonic nozzle with normal shock.As shown in these �gures, since the initial guess is nota suitable one, severe shape changes occur during itsevolution.

Wind Tunnel Supersonic Nozzle for M > 2

In order to study the FSA treatment in supersonic ow regimes with Mach number greater than 2, anideal nozzle is considered as the target shape, with thefollowing characteristics: Entrance Area = 0.14 m2,Exit Area = 0.6 m2, Length = 1.4 m, Entrance MachNumber = 1.1, Entrance Pressure = 1 bar and ExitMach Number = 2.7. A grid of 40 � 15 shown inFigure 13a is used to analyze the ideal nozzle. Thecontours of the Mach number through the nozzle andwall pressure distribution of the nozzle as the TPD areshown in Figures 13b and 14, respectively. Staring froma straight divergent duct as the initial guess, the FSAis converged to the target shape after 350 modi�cationsteps, as shown in Figure 15.

Convergent-Divergent Nozzles

The other validation test case is a convergent-divergentnozzle that includes subsonic, transonic and supersonicregimes. Its TPD is obtained from the numericalanalysis of this nozzle with a grid of 25�6 nodes, shownin Figure 16, and with the following speci�cations:Inlet Mach Number = 0.2, Inlet Pressure = 1 bar,Back Pressure=0.5 bar and Outlet Mach Number=1.5.

Figure 9. Evolution of nozzle from initial guess to the target shape with Mach number contour.


Figure 10. Initial guess and target pressure distributionfor supersonic nozzle with normal shock.

Figure 11. Wall shape modi�cation from parallel duct tosupersonic nozzle with normal shock.

Figure 12. Wall shape modi�cation from straightconvergent duct to supersonic nozzle with normal shock.

Figure 13. a) An ideal supersonic nozzle for M > 2 withits grid; b) Contour of Mach number through the Nozzle.

Figure 14. Wall pressure distribution of the idealsupersonic nozzle.

Numerical analysis of this nozzle presents a subsonicand supersonic region at the convergent and divergentparts, respectively. Starting from a constant areasection duct as the initial guess, the inverse designalgorithm converges to the convergent-divergent nozzleas the target shape after 130 evolution steps (Fig-ure 17), contrary to the high di�erence between them.Figure 18 shows the initial guess and target pressuredistribution.


Figure 15. Wall shape modi�cation from straightdivergent duct to the ideal supersonic nozzle.

Figure 16. Convergent-divergent nozzle with its grid.

Figure 17. Wall shape modi�cation from parallel duct tothe convergent-divergent nozzle with normal shock.

Figure 18. Initial guess and target pressure distributionfor convergent-divergent nozzle.

Bended Subsonic Duct with Two UnknownWalls

In all cases presented earlier in this paper and in [20],the horizontal length of the duct remains constantthrough the shape modi�cation steps. In the caseof a duct with a bend, the outlet section is notperpendicular to the x axis and, so, it is not possibleto �x the horizontal length of the duct. In such cases,instead of horizontal length, the duct centerline lengthis assumed known and remains constant during theevolution (Equation 15). In this case, the duct isasymmetric and, so, both upper and lower duct wallsare unknown. These two walls are modeled as twostrings deforming from initial guess to the target shape.During the shape modi�cation procedure, the lengthof each string is modi�ed based on the �xed centerlinelength. If the evolution of each wall is accomplishedindependently, a large number of modi�cation stepsare required to converge the design algorithm, sincein such cases the pressure is highly sensitive to thewall shape change. Considering a virtual string on theduct centerline with a constant length, and applyingthe di�erence between the upper and lower pressureto the centerline string, the convergence rate of thedesign algorithm will be increased signi�cantly. Inother words, in the case of ducts with two unknownwalls, three strings can be modeled on the upperwall, lower wall and centerline, whereby the centerlinestring, as an auxiliary string, causes the duct shapemodi�cation to speed up. The pressure di�erenceapplied to the centerline string is obtained from thefollowing equation for each link:

�p�center line = �p�upper wall ��p�lower wall: (21)


In this part, we want to redesign a duct with a 90�bend. The TPD (Figure 19) for this validation testcase is obtained from the numerical analysis of the owthrough it, with a grid of 35 � 10 nodes. Inlet Machnumber and back pressure are considered 0.5 and 1,respectively. Figure 20 illustrates the initial guess andthe evolution of the shape after 80, 200, 400 and 1000modi�cation steps. As shown in this �gure, since theinitial guess is not a suitable one, severe shape changesoccur during its evolution.

Bended Supersonic Duct with Two UnknownWalls

A supersonic 45-deg bended duct with a divergentsection was considered as a target shape, and itspressure distribution, shown in Figure 21, was obtained

Figure 19. Pressure distribution along the lower andupper wall of 90-deg bended duct.

Figure 20. Evolution of the 90-deg bended duct duringthe design process.

from numerical analysis. Inlet Mach number and inletpressure set to 2 and 1 bar, respectively, and a grid of35� 10 nodes is used for numerical analysis. Figure 22illustrates the initial guess and the evolution of theshape after 60, 150 and 400 modi�cation steps. Asshown in this �gure, since the initial guess is not asuitable one, it gets the target shape after 400 steps.

DESIGN EXAMPLES

Design of S-Shaped Ducts

Using the inverse design method, one can �nd theappropriate geometry for S-shaped ducts used as dif-fusers in the intake section of jet engines. Becauseof considerable adverse pressure gradients along theirwalls, the possibility of ow separation is very high

Figure 21. Pressure distribution along the lower andupper wall of 45-deg divergent bended duct.

Figure 22. Evolution of the 45-deg divergent bendedduct during the design process.


in such ducts. Besides, small regions may exist nearthe inlet, in which the Mach number exceeds from1 and the ow regime is supersonic. Although theadverse pressure gradients are inevitable in such ducts,one looks for S-shaped ducts without overshoot andundershoot in their wall pressure pro�les, to avoidshock waves and to reduce ow separation possibilities.

Here, we consider an arbitrary S-shaped di�useras the initial guess with an inlet area of 0.9, a lengthof 6 and an area ratio of 1.5. The upper and lowerwalls of the initial S-shaped di�user are considered astwo second order polynomials. The initial S-shapeddi�user with its grid for the numerical analysis isshown in Figure 23. The inlet Mach number and backpressure are set to 0.84 and 1.4, respectively. Theinitial pressure distribution is illustrated in Figure 24.After modifying the initial pressure distribution, theTPD is obtained in such a way that undershootsand overshoots are removed from the initial pressuredistribution and the pressure coe�cient is increased,as shown in Figure 24. After 1500 shape modi�cationsteps, the initial di�user converges to the target shape,as shown in Figure 25. Figure 26 compares the Mach

Figure 23. Initial S-shape di�user with its grid.

Figure 24. Initial pressure distribution and TPD fordesign of S-shape di�user.

Figure 25. The initial guess and target shape for S-shapedi�user.

Figure 26. Contour of Mach number through the initialand designed S-shape di�user.


number contour of the modi�ed S-shaped di�user withthat of the initial S-shaped di�user. As opposed tothe initial S-shaped di�user, the contour of the Machnumber in the modi�ed S-shape di�user decreasesfrom 0.88 to 0.45, monotonically, with no overshootand, thus, the ow remains subsonic in the entiredomain.

In order to study the grid size e�ect on thedesigned shape, the S-shape di�user, as a complicatedcase, is redesigned for three grids (24� 9, 32� 12 and40 � 15) with the same TPD in Figure 24. Figure 27compares two designed shapes. The shape related tothe 40 � 15 grid is not shown in Figure 27, because itcoincides with the shape corresponding to the 32� 12grid.

Supersonic Nozzle Design with MaximumThrust

In this part, designing a supersonic nozzle with maxi-mum thrust under an inlet and outlet speci�ed pressureis in mind. To do this, some di�erent wall pressuredistributions as TPDs are considered, and the nozzlesequivalent to these TPDs are obtained using the inversedesign code. Then, the thrust of each nozzle iscalculated from the numerical analysis. In Figure 28a,four TPDs, with respect to Equation 22, are shown.The nozzles equivalent to these TPDs are shown inFigure 28b.

P (x) = ax2 +�Pback � Pin

L+ aL

�x+ Pin: (22)

Figure 29 shows the calculated thrust versus coe�-cient (a). As seen in this �gure, the thrust is amaximum value, as the coe�cient (a) is 0.035.

Figure 27. The grid study for design of S-shape di�user.

Figure 28. a) Four TPDs for design of supersonicnozzles; b) Four designed supersonic nozzles.

Figure 29. Calculated thrust versus di�erentcoe�cients (a).


BOUNDARY CONDITION EFFECTS

In all test cases, in order to converge the designalgorithm for subsonic ow regimes, inlet Mach numberand back pressure are considered as the boundaryconditions. In the FSA method, there is no di�erencebetween the uniform and non-uniform boundary con-dition at the inlet or outlet. The authors' experienceindicated that inlet pressure and back pressure as theboundary conditions are not compatible with FSA, be-cause the free end point of the string is not compatiblewith the �xed inlet and outlet pressure. In other words,if both inlet and outlet pressures are �xed, both theinlet area and outlet area must be �xed, and the endof the string cannot be free. In order to show theboundary condition e�ect, a Michael nozzle, shown inFigure 30a is considered as the target shape. The wallpressure distribution of the Michael nozzle obtained

Figure 30. a) Geometry of Michael nozzle with its grid;b) wall pressure distribution.

from numerical analysis is illustrated in Figure 30b.Figure 31a shows that FSA converges to a Michaelnozzle with an inlet Mach boundary condition after100 iterations, whereas it diverges with a pressure inletboundary condition (Figure 31b).

CONCLUSIONS

The FSA design procedure is incorporated into anexisting Euler code with the AUSM method. The FSAturns the inverse design problem into a uid-solid in-teraction scheme that is a physical base. The method isquick converging and can e�ciently utilize ow analysiscodes as a black box. The results show that the methodcan be very promising in duct and other ow conduitdesigns for both subsonic and supersonic regimes. In

Figure 31. a) Modi�cation steps of Michael nozzle wallfrom the initial guess to the target shape versus inlet Machboundary condition, b) Divergence procedure of Michaelnozzle wall versus inlet pressure boundary condition.


the case of unsymmetrical ducts with two unknownwalls, the FSA is modi�ed to increase the convergencerate signi�cantly. Also, the results show that FSA iscompatible with inlet Mach number and back pressureboundary conditions for subsonic ow regimes.

NOMENCLATURE

a linear acceleration (m.s�2)AUSM Advection Upstream Splitting MethodCPD Current Pressure Distribution on the

wallF force vector (N)FSA Flexible String Algorithmn number of shape modi�cationsn number of linkspi pressure calculated at cell center near

the wall (Pa)Q sum of the conserved ow variables

such as �, �u, �v, �e in Euler equationsSSD Surface Shape Designt time (s)TPD Target Pressure Distributionx x position of joints (m), x coordinatey y position of joints (m), y coordinatew width of duct (m)� di�erence�s link length (m)�p�i pressure di�erence applied to each link

(Pa)�pi di�erence between TPD and CPD at

each link (Pa)� angular acceleration (rad.s�2)" convergence criterion� link angle (deg)� mass per unit length (kg.m�1)! angular velocity (rad.s�1)

Subscripts

i links indexj joints indexmax maximumx x componenty y component

Superscripts

I.G. initial guessj1 starting point of each linkj2 end point of each link

n iteration number at analysis codet+ �t related to updated geometryt related to current geometry

REFERENCES

1. Ghadak, F. \A direct design method based on theLaplace and Euler equations with application to in-ternal subsonic and supersonic ows", Ph.D Thesis,Sharif University of Technology, Aero Space Depart-ment, Iran (2005).

2. Stanitz, J.D. \Design of two-dimensional channelswith prescribed velocity distributions along the ductwalls", Technical Report 1115, Lewis Flight PropulsionLaboratory (1953).

3. Stanitz, J.D. \General design method for three-dimensional, potential ow �elds", I-Theory. TechnicalReport -3288, NASA (1980).

4. Stanitz, J.D. \A review of certain inverse methods forthe design of ducts with 2- or 3-dimensional potential ows", Proceedings of the Second International Confer-ence on Inverse Design Concepts and Optimization inEngineering Sciences (ICIDES-II), The PennsylvaniaState University, University Park, PA (Oct. 26-281987).

5. Zanetti, L. \A natural formulation for the solution oftwo-dimensional or axis-symmetric inverse problems",International Journal for Numerical Methods in Engi-neering, 22, pp. 451-463 (1986).

6. Ashra�zadeh, A., Raithby, G.D. and Stubley, G.D.\Direct design of ducts", Journal of Fluids Engineer-ing, Transaction ASME, 125, pp. 158-165 (2003).

7. Cheng, Chin-Hsiang and Wu, Chun-Yin. \An approachcombining body �tted grid generation and conjugategradient methods for shape design in heat conductionproblems", Numerical Heat Transfer, Part B, 37(1),pp. 69-83 (2000).

8. Jameson, A. \Optimal design via boundary control",Optimal Design Methods for Aeronautics, AGARD,pp. 3.1-3.33 (1994).

9. Kim, J.S. and Park, W.G. \Optimized inverse designmethod for pump impeller", Mechanics Research Com-munications, 27(4), pp. 465-473 (2000).

10. Dedoussis, V., Chaviaropoulos, P. and Papailiou, K.D.\Rotational compressible inverse design method fortwo-dimensional, internal ow con�gurations", AIAAJournal, 31(3), pp. 551-558 (1993).

11. Demeulenaere, A. and Braembussche, R. van den,\Three-dimensional inverse method for turbomachi-nary blading design", ASME Journal of Turbomachi-nary, 120(2), pp. 247-255 (1998).

12. De Vito, L. and Braembussche, R.V.D \A novel two-dimensional viscous inverse design method for turbo-machinery blading", Transactions of the ASME, 125,pp. 310-316 (2003).

Peyman

Highlight

Peyman

Highlight


13. Braembuussche, R.V.D. and Demeulenaere, A. \Threedimensional inverse method for turbomachinery blad-ing design", Journal of Turbomachinery, 120, pp. 247-255 (1998).

14. Leonard, O. and Braembussche, R. \A two-dimensional Navier stokes inverse solver for compressorand turbine blade design", Proceeding of the IMECH Epart A Journal of Power and Energy, 211, pp. 299-307(1997).

15. Henne, P.A. \An inverse transonic wing designmethod", AIAA Paper, 80-0330 (1980).

16. Volpe, G. \Inverse design of airfoil contours: Con-straints, numerical method applications", AGARD,Paper 4 (1989).

17. Barger, R.L. and Brooks, C.W. \A streamline curva-ture method for design of supercritical and subcriticalairfoils", NASA TN D-7770 (1974).

18. Campbell, R.L. and Smith, L.A. \A hybrid algorithmfor transonic airfoil and wing design", AIAA Paper,pp. 87-2552 (1987).

19. Bell, R.A. and Cedar, R.D. \An inverse method forthe aerodynamic design of three-dimensional aircraftengine nacelles", Dulikravich, pp. 405-17 (1991).

20. Malone, J.B., Narramore, J.C. and Sankar, L.N. \Ane�cient airfoil design method using the Navier-Stokesequations", AGARD, Paper 5 (1989).

21. Nili-Ahmadabadi, M., Durali, M., Hajilouy, A. andGhadak, F. \Inverse design of 2D subsonic ducts using exible string algorithm", Inverse Problems in Scienceand Engineering, 17(8), pp. 1037-1057 (2009).

22. Liou, M.-S. \Ten years in the making-AUSM-family",AIAA Paper, 2001-2521 (2001).

BIOGRAPHIES

Mahdi Nili-Ahmadabadi is a teacher of MechanicalEngineering at Isfahan University of Technology. InFeb. 2010, he graduated from Sharif University ofTechnology in Iran with a PhD in Mechanical Engi-neering. He obtained a BS and MS at Sharif Universityof Technology and Isfahan University of Technology,

respectively. He has published 5 scienti�c papers inISI journals until now. His research interests are:aerodynamic design, uid mechanics, heat transfer,turbomachines & gas turbines.

Mohammad Durali is a Professor of MechanicalEngineering at Sharif University of Technology. Hegraduated with a PhD in Aerospace Engineering fromMassachusetts Institute of Technology (MIT) in theUSA (1980). He obtained a BS and MS at SharifUniversity of Technology in Iran and MassachusettsInstitute of Technology (MIT) in the USA, respectively.He has published more than 20 scienti�c papers inISI journals and presented more than 100 papersat di�erent conferences. His research Interests are:design and automation, system dynamics & modeling,propulsion systems.

Ali Hajilouy-Benisi is an Associate Professor ofMechanical Engineering at Sharif University of Tech-nology. He graduated with a PhD in MechanicalEngineering from Imperial College, London, in theUK (1993). He obtained a BS and MS at SharifUniversity of Technology in Iran and BirminghamUniversity in the UK, respectively. He has publishedmore than 10 scienti�c papers in ISI journals andpresented more than 50 papers at di�erent confer-ences. His research Interests are: turbomachines, gasturbines, measurement, turbochargers & turbocharg-ing.

Farhad Ghadak is an Assistant Professor ofAerospace Engineering at Imam Hossein University. Hegraduated with a PhD in Aerospace Engineering fromSharif University of Technology in Iran in 2007. Heobtained a BS and MS at Tehran University and ShirazUniversity, respectively. He has published 6 scienti�cpapers in ISI journals and presented more than 10papers at di�erent conferences. His research interestsare: CFD, inverse design, wind tunnels, aerodynamics& ight control.

duct design in subsonic and supersonic

Documents