Optimal Shape Design of a Two-dimensional Asymmetric Diffsuer in Turbulent Flow

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1 THE 5 TH ASIAN COMPUTAITIONAL FLUID DYNAMICS BUSAN, KOREA, OCTOBER 7 ~ OCTOBER 30, 003 Optimal Sape Design of a Two-dimensional Asymmetric Diffsuer in Turbulent Flow Seokyun Lim and Haeceon Coi. Center for Turbulence and Flow Control Researc, Institute of Advanced Macinery and Design, Seoul National University, Seoul 5-744, Korea, lims@stokes.snu.ac.kr. Scool of Mecanical and Aerospace Engineering, Seoul National University, Seoul 5-744, Korea, coi@socrates.snu.ac.kr Corresponding autor Haeceon Coi Abstract An optimal sape of two-dimensional asymmetric diffuser in turbulent flow wit maximum pressure recovery at te exit is numerically obtained using a matematical teory. Te Reynolds number based on te bulk mean velocity and te cannel eigt at te diffuser entrance is 8,000. Te optimality condition for maximum pressure recovery is obtained to be zero skin friction along te diffuser wall. Te turbulent flow inside te diffuser is predicted using te k ε v f model and optimal sapes are obtained troug iterative procedures to satisfy te optimality condition for six different geometric constraints. For one of te optimal diffuser sapes obtained, large eddy simulation is carried out to validate te result of sape design. Keyword: optimal sape design, maximum pressure recovery, zero skin friction.. Introduction Te diffuser is a device tat converts te kinetic energy of te flow into te potential energy by broadening its eigt to decelerate te velocity and recover te pressure. However, wen te diffuser eigt increases too muc in te streamwise direction, flow separation occurs inside te diffuser, resulting in a serious pressure loss. Terefore, flow separation is an important factor tat sould be considered to enance te performance of te diffuser. Researces on modifying te diffuser wall sape ave been conducted to improve te diffuser performance. In is pioneering work, Stratford [,] teoretically obtained a pressure distribution tat maintains zero skin friction trougout te region of pressure rise inside a diffuser, and experimentally constructed a diffuser aving te teoretical pressure distribution by controlling eac segment of te diffuser wall. Maintaining zero skin friction along te diffuser wall suggests tat te diffuser sould be widened as muc as possible but preventing flow separation. Hence, te diffuser designed in tis way is expected to ave maximum pressure recovery. Traditionally, sape design metods ave primarily relied on pysical intuition. However, tese metods may need many trial-and-errors and te sape obtained is not necessarily an optimal one. Pironneau [3,4] formulated a metod of optimal sape design for fluid flow using an optimal control teory for te first time and suggested an algoritm for te optimal sape of a body wit minimum drag in low-reynolds-number laminar flow. Only recently, wit increasing computing power, te optimal sape design algoritm developed by Pironneau as been applied to some engineering problems [5-8]. In te present study, using an optimal sape design procedure, we design an optimal sape of a two-dimensional diffuser in turbulent flow wit maximum pressure recovery at te exit. Te initial diffuser sape is set to be a two-dimensional asymmetric diffuser for wic many researces ave been performed [9-]. Te Reynolds number based on te bulk mean velocity ( u b ) and te cannel eigt at te diffuser entrance ( ) is 8,000. Te turbulent flow inside te diffuser is predicted using te k ε v f turbulence model proposed by Durbin []. From te initial sape, te optimal sape of a diffuser satisfying te optimality condition for maximum pressure recovery at te exit is obtained troug te iterative solutions of te Navier-Stokes equation and its adjoint equation. 54

2 Report Documentation Page Form Approved OMB No Public reporting burden for te collection of information is estimated to average our per response, including te time for reviewing instructions, searcing existing data sources, gatering and maintaining te data needed, and completing and reviewing te collection of information. Send comments regarding tis burden estimate or any oter aspect of tis collection of information, including suggestions for reducing tis burden, to Wasington Headquarters Services, Directorate for Information Operations and Reports, 5 Jefferson Davis Higway, Suite 04, Arlington VA Respondents sould be aware tat notwitstanding any oter provision of law, no person sall be subject to a penalty for failing to comply wit a collection of information if it does not display a currently valid OMB control number.. REPORT DATE 4 APR 005. REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE Optimal Sape Design of a Two-dimensional Asymmetric Diffsuer in Turbulent Flow 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Center for Turbulence and Flow Control Researc, Institute of Advanced Macinery and Design, Seoul National University, Seoul 5-744, Korea 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 0. SPONSOR/MONITOR S ACRONYM(S). DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited. SPONSOR/MONITOR S REPORT NUMBER(S) 3. SUPPLEMENTARY NOTES See also ADM00800, Asian Computational Fluid Dynamics Conference (5t) Held in Busan, Korea on October 7-30, 003., Te original document contains color images. 4. ABSTRACT 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT UU a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified 8. NUMBER OF PAGES 8 9a. NAME OF RESPONSIBLE PERSON Standard Form 98 (Rev. 8-98) Prescribed by ANSI Std Z39-8

3 Seokyun Lim and Haeceon Coi In tis study, we design optimal sapes for six different geometric constraints (cases I-VI) suc as te lengt ( L ) and outlet eigt ( ) of te diffuser. In cases I, II and III, te diffuser lengts ( L/ ) are fixed as 0, 60 and 75, respectively, and te diffuser outlet eigt ( / ) canges freely. In cases IV, V and VI, te diffuser lengt varies as L/ = 0, 40 and 60, respectively, at a fixed diffuser outlet eigt of / = 4.7. For te optimal diffuser sape of te case V, we perform large eddy simulation to verify te sape design result.. Sape design procedure Fig. sows te scematic diagram of te initial two-dimensional asymmetric diffuser [9-]. Te eigts of te diffuser entrance and exit are and, respectively. Te diffuser lengt is L and te inlet and tail cannels, wose respective lengts are L I and L O, are connected to te diffuser entrance and exit, respectively. Γ I and Γ O are te diffuser entrance and exit boundaries. Γ M represents te diffuser wall boundary to be designed, and Γ F does te fixed diffuser wall boundary. Γ I, Γ M, Γ O and Γ constitute Γ and te inner domain enclosed by Γ is Ω. P ε in Fig. is determined by moving F eac point P on Γ in outward normal direction by te magnitude of ε ζ() s, were ζ () s is an M arbitrary function of te arc lengt s along te wall. Γ M, ε is a new diffuser wall boundary tat consists of P ε 's. Here, ε is a very small positive number and α is te opening angle of te diffuser. Turbulent mean flow inside te diffuser satisfies te continuity and Reynolds-averaged Navier- Stokes (RANS) equations: * u ii, = 0, * j i, j =, i + ν + νt i, j + ji,, j uu p [( )( u u )], were p = p/ ρ + (/3) k. Here, u i, ν, ν t, p, k and ρ are te time-averaged mean velocity, kinematic viscosity, eddy viscosity, time-averaged pressure, turbulent kinetic energy and density, respectively. Te cost function to be maximized is defined as * * J( Γ ) = ( punds+ punds), () M i i i i u b Γ I Γ O wic is te velocity-averaged pressure recovery and is commonly used as a performance indication of te diffuser. Let δ J be te variation of te cost function ( J ) due to te sape cange of te diffuser wall from ΓM to Γ (Fig. ). If we introduce te adjoint velocity ( z i ) and adjoint pressure ( r ) tat satisfy te M, ε following adjoint equations: ten δ J can be expressed as Fig.. Scematic diagram of a diffuser z ii, = 0, [( ν + ν )( z + z )] + u ( z + z ) r = 0, t i, j ji,, j j i, j ji,, i ν ui zi δj = ζ ds. u n n (4) b ΓM () (3) 55

4 Optimal sape design of a two-dimensional asymmetric diffuser in turbulent flow Wit te coice of ζ in (4) as ui zi ζ() s = ω() s /, (5) n n te variation of te cost function, δ J, is always positive, guaranteeing tat te cost function J always increases wit eac sape cange, were ω () s is a nonnegative weigting function of s (detailed derivation can be found in [8]). It is clear from (4) and (5) tat te optimality condition for maximum cost function is ui / n= 0 on Γ M, (6) i.e. zero skin friction on te diffuser wall, wic coincides wit te pysical intuition of Stratford [,]. Following is te optimal sape design procedure of a diffuser wit maximum pressure recovery at te exit: (a) assume an initial diffuser sape; (b) generate a grid inside te diffuser automatically; (c) solve te flow equations () and te adjoint equations (3); (d) obtain a new diffuser sape by moving eac point on Γ M in te outward normal direction ( n i ) by te magnitude of ε ζ() s in (5); (e) iterate (b)-(d) until te cost function converges. In te present study, te weigting function ω () s is set to be a linear function and a cosine function, respectively, for te cases of fixing te diffuser lengt and fixing te diffuser outlet eigt, suc tat te magnitude of sape cange is zero at te fixed end points. Also, ε is adjusted suc tat te magnitude of maximum sape cange does not exceed % of te diffuser outlet eigt. Te convergence criterion is selected suc tat te increase in te cost function during last ten iterations is less tan 0.5% of te total increase. 3. Numerical details Flow separation is an important factor tat affects te performance of a diffuser, and tus it is important to accurately predict te separated-flow caracteristics during te optimal sape design of a diffuser. Terefore, in tis study, we use te k ε v f turbulence model (called VF ereinafter) tat as been successfully tested for separated flows []. Te convective terms in all equations are discretized using te tird-order QUICK sceme, and te pressure-velocity coupling in () is andled wit te SIMPLER algoritm [3]. Fig. sows te computational domain and grid system for te initial diffuser sape, for wic many researces ave been performed [9-]. Te upper boundary is te flat-plate wall and te lower boundary is te diverging wall of opening angle α of about 0. Long inlet and tail cannels are connected to te diffuser entrance and exit, respectively. In practical applications, te diffuser as geometrical constraints suc as te diffuser lengt, expansion ratio, diffuser outlet eigt, etc., depending on were it is used. In te present study, two different constraints are imposed: fixing te diffuser lengt (L ) and fixing te diffuser outlet eigt ( ). In cases I, II and III, te diffuser lengts ( L/ ) are fixed as 0, 60 and 75, respectively, and te diffuser outlet eigt ( / ) canges freely. A tail cannel is connected to te end of te diffuser. In cases IV, V and VI, te diffuser outlet eigt ( / ) is fixed as 4.7 wit varying te diffuser lengt ( L/ ) as 0, 40 and 60, respectively. Te diffuser lengt, diffuser outlet eigt, number of grid points and size of te computational domain for te six cases are given in Table. y/ Fig.. Computational domain for te initial diffuser. Every fift grids in te wall-normal direction are sown in tis figure. 56

5 Seokyun Lim and Haeceon Coi Fixed L Fixed Diffuser sape Number of Computational ( L/ : / ) grid points domain size ( ) Case I 0 : free ~00 Case II 60 : free ~00 Case III 75 : free ~00 Case IV 0 : ~75 Case V 40 : ~75 Case VI 60 : ~75 Table. Sape design cases. 4. Results of te sape design In te below, we sow te results from optimal sape design for six cases described in Table. First, te results from cases I, II and III are given, and ten tose from cases IV, V and VI are described. Fig. 3 sows te optimal sapes of cases I, II and III as compared wit te initial sape. In all tree cases, te upstream parts of te optimal diffusers are nearly identical irrespective of different geometric constraints and tey seem to be little affected by te downstream geometry. Te sape of te optimal diffuser is concave up to about = 0 wit abrupt expansion around te diffuser troat and is nearly linear down to te diffuser exit ( x= L) were a tail cannel is connected. Te angle of te linear sape wit respect to te streamwise ( x ) direction is about 6. wereas te opening angle of te initial diffuser is about 0. Fig. 4 (a) sows te distribution of te skin friction coefficient along te lower wall. In te optimal sapes, te skin frictions are zero in most of te sape design region. Due to te abrupt cange in te geometry, te skin frictions near te diffuser troat are not zero even in te optimal sapes. Tere is a wide flow-separation region wit negative skin friction in te initial sape, but flow separation is Fig. 3. Optimal sapes for different constraints: -, initial sape; , optimal sape for case I;, case II;, case III. C f C p (a) Fig. 4. Skin friction and pressure coefficients along te lower wall: (a) C f ; (b) initial sape; , optimal sape for case I;, case II;, case III. (b) C p. -, 57

6 Optimal sape design of a two-dimensional asymmetric diffuser in turbulent flow completely removed in te optimal sape. Fig. 4 (b) sows te pressure distributions along te lower wall of te initial and optimal diffusers. As a result of removing flow separation, a iger pressure at te diffuser exit is obtained in te optimal sape. Wen te diffuser lengt is limited to L= 0 (case I), te pressure rises up to 0 and remains nearly constant in te downstream were flow enters te tail cannel connected to te end of te diffuser. In spite of te smaller diffuser eigt ( / = 3.4 for te optimal diffuser of case I, but / = 4.7 for te initial diffuser; see fig. 3), te pressure at = 0 in te optimal diffuser is muc iger tan tat of te initial diffuser. Te values of pressure at te exit of te tail cannel are nearly te same for bot te initial and optimal diffusers, but te pressure reaces tis value wit a sorter diffuser lengt in te optimal sape. In cases II and III, te pressure rapidly increases up to = 0 30 and ten increases very slowly in te downstream. Te pressure recovery in te optimal diffuser for case II is nearly te same as tat for case III, indicating tat a furter increase in te diffuser lengt tan tat of case II is not necessarily in terms of te pressure recovery. Fig. 5 sows te optimal sapes for cases IV, V and VI (te cases of fixed ), togeter wit tat for case III. Note tat, in cases IV, V and VI, te diffuser outlet eigt is fixed to be / = 4.7, but te diffuser lengt canges as L/ = 0, 40 and 60, respectively. Te upstream parts of te optimal diffusers, were C f is zero, are nearly identical regardless of te geometric constraint. However, te downstream parts ( C f 0 ) are very different among temselves due to fixed. Fig. 6 (a) sows te distributions of te skin friction coefficient along te lower walls of te initial and optimal diffusers for cases IV, V and VI. In case IV, L/ = 0 and / = 4.7. Due to tis severe constraint, flow separation is not removed in te optimal sape. In cases V and VI, te skin frictions are nearly zero in te sape design regions except near te diffuser troat and end. Fig. 6 (b) sows te pressure distributions along te lower walls of te initial and optimal diffusers. In all tree cases, te optimal sapes ave iger pressures at te exits of te tail cannels tan tat of te initial Fig. 5. Optimal sapes for different constraints: -, initial sape; - -, optimal sape for case III;, case IV;, case V, , case VI. C f C p (a) Fig. 6. Skin friction and pressure coefficients along te lower wall: (a) (b) C f ; (b) initial sape;, optimal sape for case IV; -, case V; , case VI. C p. -, 588

7 Seokyun Lim and Haeceon Coi C f C p (a) (b) Fig. 7. Skin friction and pressure coefficients along te upper and lower wall: (a) C p., LES; , VF. C f ; (b) sape. Te pressure at te exit of te tail cannel in case IV is lower tan tose in cases V and VI, because separation is not removed. Altoug te sape design region is extended to L/ = 60 for case VI, te exit pressures for cases V and VI are nearly te same, indicating tat, given / = 4.7, a furter increase in te diffuser lengt tan tat of case V is not necessary in terms of te pressure recovery. 5. Large eddy simulation of flow inside te optimal diffuser In te previous section, we obtained te optimal diffuser sapes for six different geometric constraints using te optimal sape design metod and VF model. In tis section, we perform a large eddy simulation (LES) of turbulent flow inside te optimal diffuser of case V, wic produced a maximum pressure recovery in a relatively sort diffuser lengt ( L/ = 40 ) wit a given diffuser outlet eigt ( / = 4.7 ), in order to confirm te sape design result from te VF model. 5. Numerical details In our LES, we use te dynamic subgrid-scale model using te least square metod [4, 5]. In calculating te model constant, te test filter is applied in te streamwise and spanwise directions using te trapezoidal rule. For time integration, te Crank-Nicolson metod is used for te viscous terms in te wall-normal direction and a tird-order Runge-Kutta metod for te oter terms. For te spatial discretization, te second-order central difference sceme is used for te streamwise and wallnormal directions and a spectral metod for te spanwise direction. All te numerical scemes are te same as tat used by Kaltenbac et al []. Te geometry of te optimal diffuser of case V is suc tat te diffuser exit eigt ( ) is 4.7 and te diffuser lengt (L ) is about 40 (see Fig. 5). Te computational domain used is =.5 50 in te streamwise direction wit te inlet cannel of lengt.5 and te tail cannel of lengt 0. Te spanwise widt is z = 4. Te numbers of grid points are in te streamwise, wall-normal and spanwise directions, respectively. Te computational domain size and grid resolution are determined based on Kaltenbac et al. [] wo successfully conducted LES for te initial diffuser sape. A convective boundary condition is imposed at te exit of te tail cannel and te periodic boundary condition is used in te spanwise direction. At te inlet of te computational domain ( x=.5 ), an unsteady fully developed turbulent cannel flow at Re = u b / ν = 8,000 ( Re τ = uτδ / ν = 500 ) is provided from a separate LES, were u τ is te wall sear velocity and δ ( = /) is te cannel alf eigt. Te computational time step is fixed to be t = 0.0 / ub. After te flow inside te diffuser reaces statistically steady state, te turbulence statistics are obtained by averaging over 4,000 time steps, during wic te flow goes troug te diffuser approximately four times. 59

8 Optimal sape design of a two-dimensional asymmetric diffuser in turbulent flow γ Fig. 8. Distribution of te reverse flow factor along te walls:, lower wall; , upper wall. 5. Results Fig. 7 sows te distributions of te skin friction and pressure coefficients along te upper and lower walls of te optimal diffuser by LES and VF. As sown, te pressure coefficients from LES and VF are in an excellent agreement, wile te skin friction coefficients are in a reasonable agreement. Wit LES, te skin friction in te sape design region is sligtly iger tan zero, but it is still very small as compared to tat of te flat plate boundary layer flow. It is also notable from te LES result tat tere exists a small separation region ( = 0 ) around te diffuser troat, wic is not observed from te VF result. Tis is similar to te finding by Kaltenbac et al. [] tat te result from LES sows a small separation bubble near te diffuser troat for te initial diffuser sape, wereas a separation bubble does not exist from te VF result. Fig. 8 sows te reverse flow factor (γ ) along te optimal diffuser walls. Te reverse flow factor is obtained as te fraction of time of flow reversal to te total time. From Fig. 8, te reverse flow factor on te lower wall increases up to 70% in te small mean separation region around x = 0, and tis factor suddenly falls off down to 40%. Te reverse flow factor is 30 40% in most of te sape design region and gradually decreases to zero. Since te mean skin friction from LES is not exactly zero but sligtly positive along te lower wall, te reverse flow factor in tis region is below tan 50%. On te upper wall, were te mean skin friction is relatively ig, γ 0%. Fig. 9 sows te contours of te instantaneous skin friction on te upper and lower walls, were te wite regions represent te negative skin friction and darker regions represent iger skin friction. In te inlet cannel region of.5 < < 0, streaky structures are clearly observed on bot te upper and lower walls. On te upper wall, tese streaks are muc attenuated after te diffuser troat ( x = 0 ) but are still observed at a downstream region. On te lower wall, tese streaks suddenly disappear at te diffuser troat ( x = 0 ), were separation bubbles appear. Tese separation bubbles are broken by ig-momentum fluids penetrating into te wall (dark spots are observed at 0 < < 5 ). Large separation bubbles are formed at a downstream region and disappear furter downstream were te skin friction becomes larger. Fig. 9. Contours of te instantaneous skin friction (wite region represent reverse flow): (a) upper wall; (b) lower wall. 60

9 Seokyun Lim and Haeceon Coi 6. Summary In tis study, we designed te optimal sape of a two-dimensional asymmetric diffuser wit maximum pressure recovery at te exit in turbulent flow, using a matematical teory. Te optimality condition for maximum pressure recovery was obtained to be zero skin friction along te diffuser wall, and optimal sapes were obtained troug iterative procedures to satisfy te optimality condition. We used k ε v f model for flow simulation. Te initial diffuser sape was set to be a two-dimensional asymmetric diffuser for wic many experimental and numerical data are available [9-], and from tis initial sape, we designed te optimal sapes for six different geometric constraints (cases I-VI). In te optimal sape obtained, te skin friction was indeed nearly zero in te sape design region, and flow separation tat appeared in te initial sape was completely removed or significantly reduced. In all te six cases, te upstream parts of te optimal diffusers were nearly identical irrespective of different geometric constraints. In te optimal diffuser, about 5~0% increase in te pressure recovery was acieved. Large eddy simulation was performed for te optimal diffuser sape of case V in order to confirm te sape design result. Te skin friction on te sape design area from large eddy simulation was sligtly iger tan zero, but was still very small as compared to tat of te flat plate boundary layer flow. Te skin friction and wall pressure from large eddy simulation were in reasonably good agreement wit tose from te k ε v f turbulence model, suggesting tat a turbulence model predicting flow separation accurately is a key element in te optimal sape design of flow systems containing separation. Acknowledgement Tis work was supported by te Creative Researc Initiatives of te Korean Ministry of Science and Tecnology. References [] Stratford, B. S., "Prediction of separation of te turbulent boundary layer," J. Fluid Mec., Vol. 5, (959), pp. -6. [] Stratford, B. S., "An experimental flow wit zero skin friction trougout its region of pressure rise," J. Fluid Mec., Vol. 5, (959), pp [3] Pironneau, O., "On optimum profiles in Stokes flow," J. Fluid Mec. Vol. 59, (973), pp [4] Pironneau, O., "On optumum design in fluid mecanics," J. Fluid Mec.,Vol. 64, (974), pp [5] Cabuk, H. and Modi, V.,"Optimum plane diffusers in laminar flow," J. Fluid Mec.,Vol. 37, (99), pp [6] Ganes, R. K.,"Te minimum drag profile in laminar flow: a numerical way," J. Fluids Engng., Vol. 6, (994), pp [7] Jameson, A., Martinelli, L. and Pierce, N.A.,"Optimum aerodynamic design using te Navier-Stokes equations," Teoret. Comput. Fluid Dynamics, Vol. 0, (998), pp [8] Zang, J., Cu, C. K. and Modi, V.,"Design of plane diffusers in turbulent flow," Inverse Problems in Engng., Vol., (995), pp [9] Obi, S., Oimuzi, H., Aoki, K. and Masuda, S.,"Experimental and computational study of turbulent separating flow in an asymmetric plane diffuser," 9t symposium on Turbulent Sear Flows, Kyoto, Japan, (993), Aug. P P [0] Buice, C. U. and Eaton, J. K.,"Experimental investigation of flow troug an asymmetric plane diffuser," Annual Researc Briefs-996, (996), pp [] Kaltenbac, H.-J., Fatica, M., Mittal, R., Lund, T. S. and Moin, P.,"Study of flow in a planar asymmetric diffuser using large-eddy simulation," J. Fluid Mec., Vol. 390, (999), pp [] Durbin, P.,"Separated flow computations wit te k-e-v model," AIAA J., Vol. 33, (995), pp [3] Patankar, S. V., "Numerical eat transfer and Fluid flow, " McGraw-Hill, (980). [4] Germano, M., Piomelli U., Moin, P. and Cabot, W. H, "A dynamic subgrid-scale eddy-viscosity model," Pys. Fluids A, Vol. 3, (99), pp [5] Lilly, D. K., "A proposed modification of te Germano subgrid scale closure metod," Pys. Fluids A, Vol. 3, (99), pp