Vortex Model for Control of DiFuser Pressure Recovery

8/3/2019 Vortex Model for Control of DiFuser Pressure Recovery

1/6

Vortex model for control of diffuser pressure recovery

Brianno D. Coller

Department of Mechanical Engineering

University of Illinois at Chicago

Chicago, IL 60607-7022

[email protected]

Abstract

The paper outlines an effort to develop moderate di-mensional computational models of the shear instabilityand subsequent highly nonlinear vortex dynamics thatoccur in a planar diffuser. The ultimate goal is to usethe model for testing and synthesis of a non-traditionalapproach to control that works by triggering instabili-ties rather than suppressing them. The models appearto capture many of the essential dynamical features,although quantitative discrepancies still need to be re-solved.

1 Introduction & Scope

A diffuser is an expanding section of a flow-carryingduct (See Figure 1a). Its purpose is to slow the meanflow in a smooth manner so that kinetic energy of theflow is converted to potential energy manifested in pres-sure rise. Except for moderate expansion angles (oftenwell under 10), though, the flow tends to separate from

the diffuser wall, shearing the flow into balled-up vortexstructures (Figure 1b) that evolve erratically, churningand reversing the flow at times. Figure 1c shows aphoto from Coller et al. [3] of a physical experiment inwhich smoke wire flow visualization clearly shows thestructures. The process generates great losses and dra-matically diminishes the diffusers ability to raise thepressure of the flow.

The dynamics one observes are the result of a shear in-stability followed by highly nonlinear fluid interactions.They are similar to phenomena observed in flow sepa-ration from general bluff bodies, and aeroelastic flutter

instabilities. While the most desirable control objectiveis to completely re-attach and re-laminarize the flow asshown in Figure 1a, the traditional control paradigm ofattempting to stabilize a system or attenuate fluctua-tions has proven to be exceedingly challenging. Mosttraditional efforts tend to require overwhelming con-trol authority, unrealistic sensing and actuation abil-ities, computational power that far surpasses currentand near future technologies, and/or require assump-tions that the systebe naturally stable or just slightly

(a) ideal, (b) actual

(c) experimental

Figure 1: The planar diffuser. Cartoons of ideal and ac-tual flow are presented in (a) and (b) respec-tively. A photo of an experiment is shown (c)in which smoke is introduced into the boundarylayer.

unstable. In more practical traditional efforts such asthose performed by Kwong and Dowling [12], the re-searchers are able to reduce peak fluctuations by 7%and spectrum power by 38% with small amplitude blow-ing and suction from the diffuser lip. However, dispitethese favorable indicators, the controller had no effecton pressure recovery.

Encouraging results, though, have recently been flow-ing from the fluid mechanics community. Researchers[14, 15, 17, 23, 16] have proposed simple open loop con-trol strategies for the diffuser by which small sinusoidalforcing is applied near the lip of the expansion eithervia mechanically actuating a small flap or by blowingand sucking fluid through a narrow slot, imparting mo-mentum perturbations on the order of 104 to 103 ofthat of the nominal flow. For appropriately chosen ac-tuation frequencies, pressure recovery is dramaticallyimproved, at times approaching the maximum theoret-ically possible [15].

The approach does not work by stabilization and at-


2/6

tenuation of fluctuations. Instead, instabilities are trig-gered in effort to favor large vortex structures that un-dergo complex interactions and evolve into configura-tions that favor the physics of pressure recovery. Un-fortunately, the control synthesis process occurs via ad-hoc, laborious trial and error. Some of the better con-trollers stumbled upon are not simple sinusoids and arenot intuitively obvious [18].

Herein, we outline the development of a moderate di-mensional computational model to capture the essen-tial natural dynamical features of the system, and itsresponse to actuator inputs.

2 Modeling

We implement a 2-D modeling technique by which thevorticity field is represented by N discrete elements ofcirculation j and positions zj . According to the vor-ticity formulation of the Navier-Stokes equation, thevorticity is simply carried with the fluid at the local

velocity and it diffuses at a rate inversely proportionalto the inverse of the Reynolds number, Re. It is an ap-proach developed by many and brought to maturity byLeonard and co-workers [1, 7, 8, 9, 19, 21]. In previousnumerical shear layer studies [4], researchers find thatthe diffusive component has little effect on global fea-tures of the roll-up and pairing dynamics. Therefore,we set Re = and ignore diffusion. Thus, circulationof the individual vortex elements remains constant, andtheir positions evolve according to the ODEs

zj = u(zj , t), j 1, 2, 3, . . . , n. (1)

Here the local velocity u(zj , t), is given by

u(zj , t) =N

k=1,=j

K(zj ,zk)k + v(zj, t), (2)

where the kernel K is the Biot-Savart law or an ana-logue for vortex elements of finite core size. The termv refers to the velocity due to the free stream and thatinduced by actuators.

2.1 Boundary Effects

For an inviscid flow the appropriate boundary condi-tion is u n = 0, where n is a vector perpendicular tothe boundary. We solve this exactly using image vor-tices and a Schwarz-Christoffel transformation. The ap-proach is a natural choice for polygonal domains and isgenerally more efficient than less accurate panel meth-ods.

We begin by finding a conformal transformation fromthe physical domain parameterized by the complex co-ordinate z = x + iy to the upper half of the complexplane Im{} 0, the transformed domain. We rep-resent the transformation compactly by (z) and the

inverse by z(). According to the Schwarz-Christoffeltheorem [2], one can construct the transformation byspecifying its derivative:

dz

d=

h

1

h1/

, (3)

where = /. In the Figure 2 the points 0 = 0,

-

1 20

h

y

x

zz1

z2

Figure 2: Transformed and physical domains for the dif-fuser problem.

1 = 1, and 2 = h1/ in the transformed domain getmapped to z0 = , z1 = 0, and z2 in the physicalplane respectively.

Solutions to the 2-D inviscid flow problem in each ofthe domains map to each other through the confor-mal transformation and its inverse. The advantage ofconformal mapping comes from the fact that boundary

conditions are easily satisfied in the transformed do-main. For each vortex of strength j at location j ,we place an image vortex of strength j outside thetransformed domain at j as shown in Figure 2, whereover-bar denotes complex conjugation. Via the map-ping, we achieve the boundary conditions in the physi-cal domain.

In typical vortex simulations reported in the literature(e.g. [20, 6, 10]) similar to this, the domain geome-tries are sufficiently simple that authors are able to ex-press (z) and z() explicitly in terms of elementaryfunctions. With such problems, it is a relatively simple

task to derive and integrate evolution equations for vor-tices in the transformed plane so they obey the properphysics in the physical domain. For the diffuser, how-ever, we are not able to explicitly integrate (3). There-fore, we must formulate the problem so that all evalua-tions and integration are performed solely in the trans-formed domain. After simulating the evolution of thevortices for a desired amount of time, we may then de-termine vortex locations in the physical plane via directnumerical integration of transformation generator (3).


3/6

Within this formulation, the evolution equations for theposition j of the jth vortex in the transformed planeis

djdt

=1

|dz/d|2

djd

ij

4(dz/d)

d2z

d2

. (4)

Derivation is too lengthy to fit within the space con-straints of this note. Here j is the complex velocitypotential within the transformed plane of the flow in-duced by the free stream, actuator, all image vortices,and all vortices except the jth vortex. dj/d, there-fore, gives the velocity resulting from all these elements.The factor |dz/d| appearing in (4) appears due to thenon-uniform scaling of the transformation as is evidentin Figure 3. In the figure, a square grid is shown in

-2000 -1000 0 1000 2000

0

1000

2000

3000

4000

-100 0 100

0

100

200

0 5 10

-5

0

5

z

(b)(a)

Figure 3: A uniform square grid in the physical plane ismapped to a set of non-uniformly spaced curvesin the transformed domain. The small box de-picts an enlargement region.

the physical domain along with its image in the trans-formed plane. The last piece in (4) is called the Routh

term. It creates rotational effects around the images ofthe vertices which are singularities in the transforma-tion.

2.2 Vorticity Generation

In the physical problem, flow separates from the dif-fuser wall at the sharp corner at the beginning of theexpansion. In this inviscid setting, we recognize thisphenomena as the inability of the fluid in the immedi-ate vicinity of the corner to sustain the near-infinite ac-celeration necessary to abruptly change direction. Theclassical modeling resolution to this issue is to imposea Kutta condition [11]. It is a constraint on the degreesof freedom of the model to ensure that pressure at thesharp corner remains finite.

The standard approach to satisfy the Kutta conditionin these types of problems is to introduce vorticity ateach time step with position and circulation chosen tocancel the velocity at the image of the convex vertex inthe transformed plane. See Figure 4. In the physical,domain, this ensures finite pressure at the corner. Onedifficulty in the approach is that the Kutta condition

zaz

a

(b)

(a)

Figure 4: Satisfaction of the Kutta condition by introduc-tion of a vortex.

is inherently non-unique. One can satisfy the Kuttacondition by introducing a weak vortex very close tothe corner, or a stronger vortex further away. Thisleads authors to introduce ad-hoc length scales [20, 6]and convection velocities for closure.

Instead, we adapt a variation developed by Giesing [5]which we find intellectually more satisfying. Instead ofintroducing a single vortex at each time step as illus-trated in Figure 4, we envision a continuous sheet ofvorticity of constant circulation density being formed

parallel to the windward side of the corner during thecourse of one time step (Figure 5). In the transformed

aazz

(a)

s

(b)

Figure 5: An alternative formulation of the Kutta condi-tion in which a continuous sheet of vorticity is

shed during a time step.

plane, then, the circulation density varies like s atleading order, where s is the distance from the image ofthe convex corner. One then chooses the magnitude ofthis distribution to cancel the flow velocity at image ofthe corner, giving a finite velocity and finite pressure inthe physical system. Conditions relating the convectionvelocity, sheet length, and strength must all be satis-fied simultaneously. The process yields a mechanicallyconsistent algorithm to introduce vortices that accountfor the separation phenomena.

2.3 Actuator Modeling

One of the goals of this modeling effort is to replicate,at least qualitatively, diffuser experiments performedby Narayanan et al. [16]. In the experimental effort, re-searchers used directed synthetic jets [13] as actuators.Schematically shown in Figure 6a, the actuator consistsof a speaker placed placed in a slotted cavity. As thespeaker diaphragm oscillates, fluid is sinusoidally forcedthrough the slot near the lip of the diffuser.


4/6

u

U

Figure 6: Schematic representation of the actuator usedin experiments performed by [16]

To incorporate such geometry with disparate lengthscales into our conformal mapping formulation of themodel causes numerical problems of crowding [22].Thus, to avoid these difficulties, yet still represent thedominant effects of vortex sheet displacement and vor-ticity generation by the actuator, we propose a muchsimpler model. As a first cut, we simply place a pointsource slightly downstream of the diffuser lip. Thesource is depicted in both the physical and transformeddomains in Figure 7.

z

z1

1 (b)

(a)

actuators

Figure 7: Schematic representation of the actuator in the physical domain (a) and transformed domain(b) in the computational model.

Narayanan et al. [16] report the peak magnitudes of the

flow issuing from the actuator slot in their experimentsalong with c, the momentum flux normalized by themomentum of the mean flow. We are able to matchthese quantities by appropriately choosing the locationof the actuator source shown in Figure 6 and the am-plitude of its sinusoidally, time-varying strength.

2.4 Vortex Merging and Removal

At each time step, the dimension of the state space in-creases by two: equations for the horizontal and verti-cal motion of each new separating vortex is introduced.To keep the problem from ballooning out of control, wealso remove vortices from the simulation once they haveconvected sufficiently far down stream that they play anegligible role in the vortex formation and pressure re-covery processes close to the separation point.

To further manage the dimension of the simulation, weallow nearby vortices sufficiently far down stream tomerge. It is a strategy explored by Shiels [21] in a vis-cous vortex simulation of flow past a circular cylinder.The resulting merged vortex has circulation equal tothe sum of the strengths of the vortices that merged

and position that coincides with their center of vortic-ity. Whether vortices are close enough to merge is de-termined by their position. Figure 8 shows, for severalstations along the duct, circles whose radii representthe relevant merge distances.

Figure 8: Merge distances for several locations throughoutthe domain.

We also remove vortices when they drift too close to thewall. Such vortices get drawn upstream due to the ve-locity induced by the image vortex. Such events do notoccur in a viscous flow as the vortex would simply beabsorbed by the boundary layer. Inviscid vortex sim-ulation efforts typically define a boundary layer thick-ness, within which vortices are simply removed [20, 6].

Figure 9: Vortices close to the wall feel a strong effectfrom its image vortex.

We have attempted this artificial boundary layer ap-proach on the diffuser with only moderate success.

Since vortex merging produces vortices with a widerange of strengths, it is difficult to choose a single lengthscale that is appropriate for all. Thus, instead of spec-ifying an ad-hoc length, we choose to remove a vortexwhen it is sufficiently close to the wall that the up-stream velocity induced by its image vortex is greaterthan free stream which tends to push the vortex downstream. The removal condition naturally scales withthe strength of the vortex.

3 Results

We run the simulation with diffuser angle = 23

in effort to compare results with previous experi-ments [16, 3]. In Figure 10, we compare nondimensionalpressure recovery Cp as a function of dimensionless forc-ing frequency or Strouhal number based on the diffuserlength L.

In Figure 10a, the experimental curves are presentedfor three different flow speeds. In all, the optimal forc-ing frequency occurs roughly at St = 0.6. Further, the


5/6

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

Stw

Cp

U = 20m/s

U = 30m/s

U = 40m/s

Cp

Cp

(b)

0 1 2 3St

-0.2

0.0

0.2

0.4 2.0St

(a)0.25

0

0

Figure 10: Pressure recovery versus dimensionless forcingfrequency for the experimental [16] system (a)and computational model (b).

L

Figure 11: Typical snapshot of vortices in uncontrolleddiffuser.

Figure 12: Two typical snapshot of vortices in controlleddiffuser at St = 0.6.

shapes of the pressure recovery curves are similar indi-cating the week influence of Reynolds number on thephysical processes that determine pressure recovery asassumed in the modeling effort. The analogous curvefor the simulation is shown in Figure 10b. Althoughthere are discrepancies in the magnitudes, the simula-tion correctly predicts the general form of the curveand the maximum pressure recovery at a frequencySt = 0.6.

In Figures 11 and 12, we show typical snapshots of thevortices produced by the numerical simulation. With-out any actuation (Fig. 11), we see that the flow is muchless organized than when we sinusoidally force the flowat its optimal frequency (Fig. 12). In Figure 12, twosuccessive snapshots are shown a short time apart. Weobserve neighboring vortices pair, creating a region ofintense circulation that draws high momentum fluid ofthe core flow close to the wall. This is the pressurerecovery mechanism we also observe in experiment asillustrated by the photograph in Figure 13 where weclearly see the smoke introduced into the core being

drawn around a region of concentrated vorticity. The

-2 0 2 4

-4

-2

0

2

-2 0 4u

2

0

-4

uncontrolled

controlled

y

Figure 13: Photo showing exchange of momentum in ex-

periment with control (left). Time - averagedvelocity profiles with and without control gen-erated by computation (right).

pairing occurs irregularly in the simulation. In an av-erage sense, the vorticity dynamics make the velocityprofile at the dotted line in Figures 11 and 12 more fullas illustrated in Figure 13. Simple control volume tech-niques developed in [16] and reported in [3] illustratehow fuller profiles improve pressure recovery character-istics.

4 Conclusions

The simulations presented herein contain on the orderof 700 vortices. Even with this coarse representationand other modeling assumptions, initial results indicatethat our vortex model is capable of capturing many ofthe essential features of this complex dynamical sys-tem. There is still room for improvement, however, inquantitative results. The models pressure recovery pre-


6/6

dictions are too low, and the negative values of Cp donot make physical sense. In closer examination of thevortex simulations, it appears as though the anomaly isdue to vortices close to the wall artificially being carriedupstream by their image vortices. While the new mod-eling approaches outlined in Section 2.4 has improvedquantitative agreement substantially over that reportedin [3] close examination of the data indicates that oneshould modify the artificial boundary layer to account

for concentrated clusters of vorticity in addition to in-dividual vortices.

Nonetheless, the model is very close to being a use-ful tool to efficiently test and synthesize new controllaws without resorting to experiment or typical detailednumerical simulations which take orders of magnitudelonger to run.

Acknowledgement: I wish to acknowledge fruitfuldiscussions with Bernd Noack, Satish Narayanan, An-drezj Banaszuk, and Alex Khibnik.

References

[1] C. Chang and R. Chern. A numerical study offlow around an impulsively started circular cylinder bya deterministic vortex method. Journal of Fluid Me-chanics, 233:243, 1991.

[2] R.V. Churchill and J.W. Brown. Complex Vari-ables and Applications. McGraw Hill, 1990.

[3] B.D. Coller, B.R. Noack, S. Narayanan, A. Ba-naszuk, and A.I. Khibnik. Reduced-basis model for ac-tive separation control in a planar diffuser flow. AIAAPaper 20002562, 2000.

[4] A.F. Ghoniem and K.K Ng. Numerical study ofthe dynamics of a forced shear layer. Physics of Fluids,30:706 721, 1987.

[5] J.P. Giesing. Vorticity and Kutta condition forunsteady multienergy flows. Journal of Applied Me-chanics, 37:608615, 1969.

[6] M. Kiya, K. Sasaki, and M. Arie. Discrete-vortexsimulation of a turbulent separation bubble. Journal ofFluid Mechanics, 120:219244, 1982.

[7] P. Koumoutsakos. Direct Numerical Simulationsof Unsteady Separated Flows Using Vortex Methods.PhD thesis, California Institute of Technology, 1993.

[8] P. Koumoutsakos and A. Leonard. High-resolution simulations of the flow around an impulsivelystarted cylinder using vortex methods. Journal of FluidMechanics, 296:1, 1995.

[9] P. Koumoutsakos and D. Shiels. Simulations ofthe viscous flow normal to an impulsively started anduniformly accelerated flat plate. Journal of Fluid Me-chanics, 328:177, 1996.

[10] R. Krasny. Vortex sheet computations: roll-up,wakes, separation. Lectures in Applied Mathematics,28:385402, 1991.

[11] A.M. Kuethe and Y.C. Chow. Foundations ofAerodynamics: Bases of Aerodynamic Design. Wiley,1986.

[12] A.H.M. Kwong and A.P. Dowling. Activeboundary-layer control in diffusers. AIAA Journal,32:24092414, 1994.

[13] D.C. McCormick. Boundary layer separationcontrol with directed synthetic jets. In 39th AIAAAerospace Sciences Meeting and Exhibit, 1999.

[14] Jr. McKinzie, D.J. Turbulent boundary layer sep-aration over a rearward facing ramp and its controlthrough mechanical excitation. AIAA Paper, 91-0253,1991.

[15] Jr. McKinzie, D.J. Delay of turbulent boundarylayer detachment by mechanical excitation: Applica-tion to rearward-facing ramp. Technical Report 3541,NASA Technical Paper, 1996.

[16] S. Narayanan, B.R. Noack, A. Banaszuk, andA.I. Khibnik. Dynamic flow separation control ina low-speed planar diffuser. Technical Report R99-1.910.9901-4.1, United Technologies Research Center,1999.

[17] B. Nishri and I. Wygnanski. Effects of periodicexcitation on turbulent flow separation from a flap.AIAA Journal, 36(4):547556, 1998.

[18] D.E. Parekh and Glezer A. Avia: Adaptive vir-tual aerosurface. AIAA Paper 20002474, 2000.

[19] F. Pepin. Simulation of the Flow Past an Impul-sively Started Cylinder Using a Discrete Vortex Method.

PhD thesis, California Institute of Technology, 1990.[20] T. Sarpkaya. An inviscid model of two-dimensional vortex shedding for transient and asymp-totically steady separated flow over an inclined plate.Journal of Fluid Mechanics, 68:109128, 1975.

[21] D. Shiels. Simulation of Controlled Bluff BodyFlow with a Viscous Vortex Method. PhD thesis, Cali-fornia Institute of Technology, 1998.

[22] L.N. Trefethen. Numerical computation of theschwarz-christoffel transformation. SIAM J. Sci. Stat.Comput., 1, 1980.

[23] H. Viets, M. Piatt, and M. Ball. Forced vorticesnear a wall. AIAA Paper 810256, 1981.

Vortex Model for Control of DiFuser Pressure Recovery

Documents

Transcript of Vortex Model for Control of DiFuser Pressure Recovery