# Detecting fluid flows with bioinspired hair sensors

TABLE OF CONTENTS Page 1 General Introduction 1 2 The Detection of Unsteady FlowSeparation with Bioinspired Hair-Cell Sensors 5 2.1 Abstract..................................6 2.2 Introduction................................6 2.3 Hair-Cell Model..............................9 2.3.1 Discretization of Sensor Model..................11 2.4 The Problem Deﬁnition..........................13 2.5 Results and Discussion..........................16 2.6 Linear Algebraic Sensor Model......................21 2.7 Summary.................................25 3 Mathematical Modeling of Biologically Inspired Hair Receptor Arrays in Lam- inar Unsteady Flow Separation 26 3.1 Abstract..................................27 3.2 Introduction................................28 3.3 Mathematical Modeling of the Hair/Fluid Problem..........30 3.3.1 Flow Over an Impulsively Started Cylinder From Rest....32 3.3.2 Viscoelastic Hair Forced by a Viscous Flow...........33 3.4 Simulation Details............................37 3.4.1 Finite Element Solution of the Fluid Model..........37 3.4.2 Finite Element Solution of the Hair Model...........37 3.5 Simulation Results and Discussion....................39 3.5.1 Simulation of the Flow Over an Impulsively Started Cylinder.39 3.5.2 Dynamic Response and Output of Hair at 15 ◦ .........40 3.5.3 Detection of Unsteady Flow Separation with Hair Array Moments 44 3.6 Summary.................................46 4 Boundary Layer Detection with Hair Sensors 48 4.1 Abstract..................................49 4.2 Introduction................................49 4.3 Hair Sensor Model............................52 4.3.1 Nondimensional Form of Hair Model..............56 4.3.2 Boundary Layer Model......................57 4.4 The Optimal Hair Length for Boundary Layer Detection.......59 4.4.1 Hairs with Uniform Cross-Section................60 4.4.2 Hairs with Linearly Tapered Cross-Section...........62 4.4.3 Comparison of Optimal Hair Lengths with Biological Data..65 4.5 Summary.................................68

TABLE OF CONTENTS (Continued) Page 5 A Snapshot Algorithm for Linear Feedback Flow Control Design 70 5.1 Abstract..................................71 5.2 Introduction................................71 5.3 Problem Description...........................73 5.3.1 An Abstract Formulation.....................75 5.3.2 The Control Problem.......................77 5.4 Computational Approach.........................77 5.4.1 A Snapshot Algorithm for Feedback Gains...........78 5.4.2 Implementation Details for the Stokes Control Problem....82 5.5 Numerical Results.............................85 5.6 Summary.................................87 6 Addendum:Observer Design for an Unsteady Stokes-Type Flow using Bioin- spired Hair Sensor Arrays 88 6.1 Computational Approach.........................91 6.1.1 The Snapshot Algorithm for Observer Functional Gains....91 6.1.2 Implementation Details for the Stokes Observer Problem...95 6.2 Numerical Results.............................96 6.2.1 Summary.............................104 8 Conclusions 106

LIST OF FIGURES Figure Page 2.1 Scanning electron micrograph of hair cells on the wing of the grey- headed ﬂying-fox.............................7 2.2 Photo of polymer artiﬁcial hair cell sensors...............8 2.3 Fluid domain for unsteady ﬂuid separation simulations performed in this analysis.................................13 2.4 Illustration of hair-cell numbering scheme where 179 sensors were placed at n =1,2...,179 degrees measured fromthe horizontal plane (sensors not shown to scale)............................15 2.5 Nondimensional velocity magnitude snapshot at t = 0.50 s with the ﬂow attached everywhere.........................17 2.6 Nondimensional velocity magnitude snapshot at t = 2.00 s with re- versed ﬂow and point of zero wall shear at 66.5 ◦ ............17 2.7 Nondimensional velocity magnitude snapshot at t = 3.50 s with re- versed ﬂow and point of zero wall shear at 68.3 ◦ ............17 2.8 Nondimensional velocity magnitude snapshot at t = 5.00 s with re- versed ﬂow and point of zero wall shear at 68.6 ◦ ............17 2.9 Image plot of the 179 hair-cell sensor array response to the separating ﬂow simulation plotted against sensor position number and time showing 18 2.10 Vector velocity and velocity magnitude for t = 3.50 s near the cylinder wall spanning approximately 30 ◦ to 70 ◦ and showing the presence of a clockwise rotating eddy between the 35 ◦ and 50 ◦ marks that is trapped by the large downstream counterclockwise eddy............20 2.11 Load intensity,g(t,ξ),acting on left sensor causing moment,M(t),and equivalent point-load,F t (t),acting on the right sensor at the center of mass of g(t,ξ), ¯ ξ(t),to produce the equivalent moment M(t).....21 2.12 Sensor array response computed with linear algebraic model.....24 2.13 Sensor array response from ﬁnite element simulation..........24 3.1 Scanning electron micrograph of hair receptors on the wing membrane of Pteropus poliocephalis (the grey-headed ﬂying-fox).........28

LIST OF FIGURES (Continued) Figure Page 3.2 Photo of polymer artiﬁcial hair sensor (left) and force sensitive resistor (FSR) at base of polymer AHC (right).................29 3.3 Illustration of hair sensor array position and numbering where sensors are placed at n = 1 ◦ ,2 ◦ ...,179 ◦ measured from the horizontal plane (sensor length and diameter are not shown to scale)..........31 3.4 Fluid domain for unsteady ﬂuid separation simulations performed in this analysis.................................32 3.5 Nonuniform incident ﬂow velocity (a) and corresponding freebody di- agram (b) of a hair............................35 3.6 Nondimensional velocity magnitude snapshot at t ∗ = 0.064 with the ﬂow attached everywhere.........................40 3.7 Nondimensional velocity magnitude snapshot at t ∗ = 0.40 with point of zero wall shear stress at 59.9 ◦ .....................40 3.8 Nondimensional velocity magnitude snapshot at t ∗ = 1.19 with point of zero wall shear stress at 73.7 ◦ .....................40 3.9 Relative ﬂow velocity for hair 15.....................41 3.10 Load intensity acting on hair 15.....................41 3.11 Deﬂection of hair 15...........................42 3.12 Output of hair sensor 15.........................42 3.13 Tip deﬂection of sensor 15 versus time for various τ..........43 3.14 Tip velocity of sensor 15 versus time for various τ...........44 3.15 Moment output of sensor 15 versus time for various τ.........44 3.16 Hair array output to ﬂow over an impulsively started cylinder simulation 45 3.17 Vector velocity at t ∗ = 1.50 near the cylinder wall showing the counter rotating eddies detected by the hair array between 33 ◦ and 74 ◦ ....46 4.1 Scanning electron micrograph of hair receptors on the wing membrane of Pteropus poliocephalis (the grey-headed ﬂying-fox).........50

LIST OF FIGURES (Continued) Figure Page 4.2 Nonuniform ﬂow velocity proﬁle incident on hair receptor (left) and corresponding free body diagram of hair (right)............53 4.3 Solutions of the Falkner-Skan equation ranging from separation (β = −0.199) to plane stagnation (β = 1.0)..................58 4.4 M ∗ as a function of ∗ and H for a hair sensor with uniform cross section 61 4.5 F ∗ as a function of ∗ and H for a hair sensor with uniform cross section 61 4.6 Sensitivity of hair,S M ∗ and S F ∗ ,with a uniformdiameter as a function of ∗ ....................................62 4.7 M ∗ as a function of ∗ and H for a hair sensor with tapered cross-section 63 4.8 F ∗ as a function of ∗ and H for a hair sensor with tapered cross-section 63 4.9 Sensitivity of hair,S M ∗ and S F ∗ ,with a linearly tapered diameter as a function of ∗ ...............................63 5.1 Functional gain for horizontal velocity,k 1 ................86 5.2 Functional gain for vertical velocity,k 2 .................86 6.1 Illustration of the ﬂow observer problem with hair sensor arrays...88 6.2 Flow observer problem schematic showing estimation regions and sen- sor arrays.................................97 6.3 Observer functional gain for horizontal velocity measurement of hori- zontally mounted hair sensor array,g 11 .................100 6.4 Observer functional gain for vertical velocity measurement of horizon- tally mounted hair sensor array,g 12 ...................101 6.5 Observer functional gain for horizontal velocity measurement of hori- zontally mounted hair sensor array,g 21 .................101 6.6 Observer functional gain for vertical velocity measurement of horizon- tally mounted hair sensor array,g 22 ...................102 6.7 Evolution of state estimate error of horizontal velocity in R 1 with and without hair sensors...........................104

LIST OF FIGURES (Continued) Figure Page 6.8 Evolution of state estimate error of vertical velocity in R 2 with and without hair sensors...........................104

LIST OF TABLES Table Page 2.1 Geometric and material parameters of the hair-cell sensor model (2.2) 16 3.1 Geometric and material parameters used in the simulation of each hair 39 4.1 Summary of optimal relative hair lengths determined herein for hairs with uniform and linearly tapered cross-section............64 4.2 Measured bat wing hair receptor lengths and optimal hair lengths com- puted from bat wing and ﬂight measurements.............66 5.1 Lyapunov iteration number and time steps for convergence of corre- sponding Lyapunov solutions.......................86 6.1 Lyapunov iteration number and time steps for convergence of corre- sponding Lyapunov solutions.......................100

Detecting Fluid Flows with Bioinspired Hair Sensors 1 GENERAL INTRODUCTION The eﬀective use of artiﬁcial hair sensors (AHS) for the detection and feedback of information related to aerodynamically or hydrodynamically important ﬂuid ﬂows will require an understanding of the relationships between mechanical quantities of hair-like structures and the particular phenomena or physical quantities speciﬁc to the ﬂows of interest.In this collection of manuscripts,we investigate hair-like struc- tures for the detection and feedback of information related to aerodynamically and hydrodynamically important ﬂows.Speciﬁcally,we will 1) determine that an AHS array can provide a space and time accurate representation of laminar unsteady ﬂow separation,2) characterize how hairs physically respond to unsteady viscous incom- pressible ﬂows and how the mechanical moment and shear force at the base of the hair is related to local ﬂow velocity,3) determine how hair geometry inﬂuences the hair sensitivity in boundary layer ﬂows,4) develop a tractable mathematical model of hair-like structures for ﬂow control applications,and 5) provide evidence that limited wall measurements from hair sensor arrays can eﬀectively estimate the regions of the ﬂow velocity ﬁeld through model-based observer design. Many animals use hair-like structures to detect their ﬂow environments.For example,bats exhibit super-maneuverability in low-Reynolds number regimes of ﬂight and acrobatic-like behavior when landing.While this is likely largely an outcome of the bat’s articulated wing structure,bats also possess distributed arrays of hair receptors growing from their wing surfaces.It has been hypothesized that the hair receptor arrays provide instantaneous feedback on the airﬂow environment over the

2 wing [1,2,3] allowing the bat to adjust its kinematics or wing shape during ﬂight. Cartilaginous and bony ﬁshes also use mechanosensor arrays of hair-like structures known as the lateral line.By detecting changes in ﬂow environment surrounding the ﬁsh,the lateral line has been implicated in prey detection and tracking,collective schooling behavior,and maintaining position and orientation in strong currents [4,5]. Other examples of biological hair receptors may be found on the legs of crickets [6,7] and spiders [8,9]. Inspired by the biological hair receptor,ﬂow feedback provided by AHS is one potential means of detecting aerodynamic or hydrodynamic forces on a body.For example,micro-air-vehicles (MAV) operate in low-Reynolds number ﬂight regimes (Re ∼ 10 5 ) with inherent ﬂow unsteadiness and ﬂight stability issues caused by gusts of wind or separation bubbles (see [10] and the references therein).One potential solution to the challenges associated with low-Reynolds number ﬂight is a closed-loop ﬂow control system integrated into the MAV and designed to mitigate the eﬀects of such destabilizing ﬂows.Airﬂow feedback to controller would be likely provided by measurements of the ﬂow ﬁeld made from the MAV surface.For this purpose, a suite of surface mounted hair sensor arrays is one potential means of ﬂow detec- tion.However,the successful application of AHS requires an understanding of the relationship between their local ﬂow environment and physical response.This is a general requirement for any AHS application and is addressed here by the 5 research objectives listed above. The article in Chapter 2 [11] contributes to the ﬁrst research objective:the me- chanical characterization of a hair sensor array in unsteady ﬂow separation.Here, each hair sensor is modeled as a viscoelastic Euler-Bernoulli beam and coupled to the ﬂow with empirical drag coeﬃcients for cylinders in cross-ﬂow.With ﬁnite element simulations of the hair-ﬂuid model we found that the moment at the base of the hair

3 (taken as the hair output signal) provided a time and space accurate representation of the phenomena associated with unsteady ﬂow separation.A linear algebraic hair sen- sor model was then derived and shown to provide an output similar to the viscoelastic hair simulations. Chapter 3 [12] contributes to the ﬁrst and second research objectives.In addition to further characterization of an AHS array in unsteady ﬂow separation,the eﬀect of the hair material properties on its dynamic response and how the hair dynamics inﬂuence the resultant moment at the base of the hair is studied.Here,we show that the hair output is independent of the hair dynamics and dominated by the surface forces from the viscous ﬂow.This result indicates that inertial forces of the hair,and thus the hair dynamics,may be neglected in modeling the relationship between the ﬂow velocity incident on the hair and the resultant mechanical response at its base. These results also justiﬁed the similarity between the linear algebraic hair model and ﬁnite element simulations presented in Chapter 2. The third objective,the eﬀect of hair geometry on output sensitivity,is presented in the manuscript contained in Chapter 4.Based on the physical analysis contained in Chapter 3,a simpliﬁed quasi-steady model of the hair is developed.The relative hair length to any boundary layer ﬂow that maximizes hair output sensitivity is determined.The range of computed optimal hair lengths are shown to be in close agreement with measured biological values.These results support the hypothesis that bats use hair sensors for boundary layer detection and provide geometric guidelines for artiﬁcial hair sensor design. The article in Chapter 5 contains a proof-of-principle study for the ﬁfth objective: ﬂow observer design with hair sensor arrays.Observer (and control) design for ﬂow problems (in general distributed parameter systems) is not without its own set of theoretical and computational challenges.Here we use a new algorithm [13,14]

4 to compute linear quadratic control laws for an unsteady Stokes-type ﬂow.This methodology is successfully applied to an observer design with hair sensor arrays as an addendum to this thesis in Chapter 6.This work is concluded in Chapter 8 with a summary of the results and conclusions of each manuscript.

5 THE DETECTION OF UNSTEADY FLOWSEPARATION WITH BIOINSPIRED HAIR-CELL SENSORS B.T.Dickinson,J.R.Singler,B.A.Batten Proceedings of the 26th AIAA Aerodynamic Measurement Technology and Ground Testing Conference,2008,AIAA paper number 2008-3937

6 2.1 Abstract Biologists hypothesize that thousands of micro-scale hairs found on bat wings function as a network of air-ﬂow sensors as part of a biological feedback ﬂow control loop.In this work,we investigate hair-cell sen- sors as a means of detecting ﬂow features in an unsteady separating ﬂow over a cylinder.Individual hair-cell sensors were modeled using an Euler- Bernoulli beam equation forced by the ﬂuid ﬂow.When multiple sensor simulations are combined into an array of hair-cells,the response is shown to detect the onset and span of ﬂow reversal,the upstream movement of the point of zero wall shear-stress,and the formation and growth of eddies near the wall of a cylinder.A linear algebraic hair-cell model,written as a function of the ﬂow velocity,is also derived and shown to capture the same features as the hair-cell array simulation. 2.2 Introduction Numerous reconnaissance and surveillance applications exist for autonomous micro- air-vehicles (MAV).However,the utility of the MAV is limited due in part to poor resistance to adverse pressure gradients and the formation of laminar separation bub- bles that occur in their low-Reynolds number ﬂight regimes.To mitigate the eﬀects of such destabilizing ﬂow phenomena,some researchers have sought to design closed- loop ﬂow controllers,which when implemented on an MAV will require novel ﬂow sensors due to payload and power limitations.In this work,we investigate a means of ﬂow detection by drawing biological inspiration from the extraordinarily complex and extremely maneuverable ﬂight of the bat. Biologists have recently provided new evidence to suggest that thousands of hair-

7 cells (see Figure 2.1),scattered across the bat wing surface are actually a distributed sensing network that provides boundary layer feedback as part of a biological ﬂow control loop [2].The hair-cells shown in Figure 2.1 belong to the grey-headed ﬂying- Figure 2.1:Scanning electron micrograph of hair cells on the wing of the grey-headed ﬂying-fox (Reproduced by permission of CSIRO PUBLISHING,from the Australian Journal of Zoology vol.42(2):215-231 (GV Crowley and LS Hall).Copyright CSIRO 1994.http://www.publish.csiro.au/nid/91/issue/2300.htm) fox and are thought to be air-ﬂow sensors [1].The hairs are on the order of 1 mm tall,and protrude from dome structures which contain touch-sensitive cells.Apart from the bat,hair-cell arrays are also found in cartilaginous and bony ﬁshes,where they are implicated in prey detection and tracking,collective schooling behavior,and maintaining position and orientation in strong currents [4].Interestingly,hair-cells are also thought to play a role in boundary layer detection for locomotion control in ﬁsh.Other examples of hair cell use include movement sensitive touch detection in spiders [15] and sound vibration detection in mammals. For controller implementation,new micro-electro-mechanical manufacturing tech- nology is changing the hair-cell from a biological curiosity into an available sensor. Inspired by the biological hair-cell,the Micro Nano Technology Research (MNTR)

8 group 1 has designed,manufactured,and tested high sensitivity artiﬁcial hair-cell (AHC) sensors [16,17,18].One AHC design is composed of an all-polymer hair attached to a force sensitive resistor base,as shown in Figure 2.2.Additionally,the Figure 2.2:Photo of polymer artiﬁcial hair cell sensor (left) and force sensitive resistor (FSR) at base of polymer AHC (right).Figures courtesy of C.Liu and group,MNTR lab,Northwestern University AHC may be manufactured as small as 10 µm in diameter. 2 The design of a model-based controller with an array of hair-cell sensors will require an accurate model of the sensor array.In our previous work [19],a physical model for an individual AHC was developed,based on those manufactured by the MNTRgroup [17].Here,we will investigate the response of an AHCarray to unsteady ﬂow separation and derive a linear algebraic model for a hair-cell array for application in linear control designs. In the next section,we develop our model of the hair-cell sensor and state its assumptions.The ﬂow and sensor array problem statement is presented in Section 2.4,followed by the results of the ﬂow and hair-cell simulations in Section 2.5.In Section 2.6,a linear sensor model is derived and its response is compared to our sensor array simulations.Finally,we summarize our ﬁndings and outline avenues of future research in Section 2.7. 1 F ormerly at the University of Illinois at Urbana Champaign (UIUC),now at Northwestern University 2 Private communication with Chang Liu,Northwestern University

9 2.3 Hair-Cell Model When placed in an air ﬂow,the biological hair-cell is subject to a net drag force that acts normal to its length,causing the output of bioelectrical signals from the hair- cell dome.With similar function,artiﬁcial hair-cell (AHC) designs have integrated micro-electro-mechanical mechanisms in their base [16,17,18].In this work,we take the sensor’s output as the bending moment at its base and note that future mod- els of particular AHC designs may require additional modeling of electromechanical mechanisms. To describe the relationship between the external ﬂow around a hair-cell and the resulting bending moment,we use an Euler-Bernoulli beam equation coupled to a constitutive drag force equation.Viscoelastic material damping is also included with the Kelvin-Voigt material model.This leads to the following partial diﬀerential equation to describe the dynamics of each hair sensor ρ s Ar tt (t,ξ) +γIr tξξξξ (t,ξ) +EIr ξξξξ (t,ξ) = g(t,ξ),0 < ξ < L,0 < t < T,(2.1) with boundary conditions r(t,0) = 0,r ξ (t,0) = 0, EIr ξξ (t,L) +γIr tξξ (t,L) = 0,EIr ξξξ (t,L) +γIr tξξξ (t,L) = 0, 0 < t < T, and initial condition r(0,ξ) = r 0 ,0 ≤ ξ ≤ L, where r(t,ξ) denotes the deﬂection of the hair at time t and position ξ,L is the hair’s length,d is its diameter,ρ s is the density of the hair-cell,A is its cross-sectional area,E is Young’s modulus of the hair material,I is the moment of inertia,γ is the

10 Kelvin-Voigt coeﬃcient of material damping,and g(t,ξ) is the load intensity,with units of force per unit length.Additionally,the subscripts (·) ξ and (·) t denote partial derivatives. To express the load intensity,g(t,ξ),due to the ﬂow we use the drag force equation g(t,ξ) = sgn(u n (t,ξ)) 1 2 C f (u n ) ρ a du n (t,ξ) 2 ,(2.2) where u n (t,ξ) is the ﬂuid velocity ﬁeld projected normal to the length of the hair, sgn(u n ) accounts for the direction of u n (t,ξ),ρ a is the density of air,and C f is the drag coeﬃcient for a cylinder in cross ﬂow,which was computed pointwise along the length of the sensor to account for the nonuniform velocity boundary layer. A relationship between C f and u n was determined by ﬁtting a ﬁrst-order poly- nomial to the logarithm of empirical drag coeﬃcients for an inﬁnite cylinder versus Reynolds number as lnC f (u n ) ≈ −0.67 lnRe

(u n ) +2.51,(2.3) where the Re(u n ) is a function of u n (t,ξ) and based on the hair’s diameter as Re

(u n ) = u n (t,ξ) d ν .(2.4) The drag coeﬃcient equation (3.7) is an accurate approximation for Re

< 7. Finally,the moment at the base of the hair-cell may be computed as M(t) = EIr ξξ (t,0) +γIr tξξ (t,0).(2.5) In constructing the hair-cell model (2.2),we neglect any forces on the sensor from

11 ﬂow phenomena on its free end,such as recirculation at the tip.Additionally,we do not account for any ﬂow eﬀects at the base of the sensor,such as a horseshoe vortex. By examining orders of magnitude in the Reynolds number expression (2.4),we ﬁnd that for air passing over a hair-cell with L = 1 mm and d = 10 µm (the dimensions used herein),| u n (t,ξ) | ∼ Re

.Thus,for u n << 1,we have Re

<< 1 and the ﬂow over the sensor will be very smooth.By limiting the sensor’s height to 1 mm,so that it remains submerged within the boundary layer of our simulations,we ensure that u n (t,ξ) << 1 m/s.Additionally,the immersion of the sensor in the boundary layer may help ensure its sensitivity to the boundary layer ﬂow while the hair’s protrusion into the freestream may saturate its response. We also assume the eﬀect of the sensor on the surrounding ﬂow ﬁeld is negligible. Thus,we couple the sensor to the ﬂow through the load intensity,g(t,ξ),but do not couple the ﬂow to the sensor in the Navier-Stokes equations.Although,the true extent of the sensor’s eﬀect on the ﬂow is unknown,preliminary wind tunnel experiments performed at Oregon State University with hair-cells mounted on the surface of micro-air-vehicle wings have supported this assumption. 3 Finally,we assume that the velocity of any point on the sensor is much less than the ﬂow velocity acting at that point,that is r ξ (t,ξ) << u n (t,ξ) for 0 ≤ ξ ≤ L and 0 ≤ t ≤ T.To this end,we do not compute a relative normal ﬂow velocity due to the sensor’s motion. 2.3.1 Discretization of Sensor Model In this section,we describe the discretization of the hair-cell sensor model (2.1) with the ﬁnite element method.To compute approximate solutions,r(·,·),we multiply 3 Private communication with Dan Morse,Oregon State University

12 the sensor model (2.1) by a test function φ(·) and integrate by parts twice to look for solutions r(·,·) ∈ L 2 (0,T;X(0,L)) such that ρA(r tt ,φ) +γI (r tξξ ,φ ξξ ) +EI (r ξξ ,φ ξξ ) −(g,φ) = 0,∀φ ∈ X(0,L),(2.6) where X(0,L) = {φ ∈ H 2 (0,L) | φ(0),φ ξ (0) = 0},and (f,g) =

L 0 f(x) g(x) dx denotes the standard L 2 inner product.In the ﬁnite element discretization of (3.12), we look for approximate solutions r h ∈ X h such that ρA

r h tt ,φ h

+γI

r h tξξ ,φ h ξξ

+EI

r h ξξ ,φ h ξξ

−

g,φ h

= 0,∀φ h ∈ X h ,(2.7) where X h ⊂ X(0,L) is a ﬁnite dimensional space spanned by cubic B-splines on a grid deﬁned over (0,L) and r(t,ξ) ≈ r h (t,ξ) = N

i=1 R i (t)φ i (ξ). Substituting r h into the ﬁnite element form (3.13) gives the second-order system of ordinary diﬀerential equations (ODE) MR

+AR

+C R = F(t) (2.8) to be solved for R = [R 1 (t)...R N (t)],where (·)

denotes a time derivative.The ODE system (3.14) was set up as a system of ﬁrst order equations in [R,R

] and solved using a backward diﬀerentiation formula (BDF) method.Implementation details are presented in Section 2.4.

13 2.4 The Problem Deﬁnition The ﬂow problem described here follows the impulsively started cylinder problem described by Gresho and Sani [20] (pages 794-845).Figure 2.3 illustrates the artiﬁcial ﬂow domain,Ω,for a cylinder in cross ﬂow used our simulations.Let u(t,x) = Figure 2.3:Fluid domain for unsteady ﬂuid separation simulations performed in this analysis. [u(t,x,y),v(t,x,y)] denote the two-dimensional velocity ﬁeld and p(t,x) denote the pressure ﬁeld which describe the ﬂuid dynamics in Ω,modeled by the nondimensional viscous,incompressible Navier-Stokes equations u t +u · ∇u = ∇p + 1 Re ∇ 2 u ∇· u = 0 (2.9) with the following initial and boundary conditions, u = (1 −e −λt ),v = 0 on Γ i × (0,T], −p n + 1 Re ∂u ∂n = 0 on Γ o × (0,T], ∂u ∂n = 0,v = 0 on Γ t,b × (0,T], u = 0 on Γ c × (0,T], u(0,x) = 0 in Ω,