An Overview of Computational Fluid Structure Interaction: Methods and Applications
AAn Overview of Computational Fluid StructureInteraction: Methods and Applications
Sumant R. Morab
Doctoral StudentDepartment of Mechanical EngineeringIndian Institute of Technology, BombayEmail: [email protected]
Atul Sharma ∗ ProfessorDepartment of Mechanical EngineeringIndian Institute of Technology, BombayEmail: [email protected]
Over the past few decades, there has been a rapidimprovement in computational power as well astechniques to simulate the real world phenomenonwhich has enabled us to understand the physics anddevelop new systems which outperform the existingones. In the domain of multi-physics problems,fluid and structure interactions have been studiedand various numerical methods are introduced tosolve them. An extensive review of most commonlyemployed numerical techniques to solve fluid struc-ture interaction(FSI) problems has been done inthis article. It also becomes utmost important tounderstand the applicability of each method andhence a critical reasoning for usage of differentmethods has been provided. Application domainsare presented which range from energy harvestingprocesses to simulation of human vocal cords andmovement of bolus(food) inside esophagus. Suitablenumerical methods applied for each application havealso been discussed. Numerical methods developedso far are classified in particular groups based onthe discretization, the manner in which coupling isperformed and on the basis of meshing(division ofentire domain into small blocks inside which varia-tion of a variable is approximated). Challenges andinstabilities posed by presently available numericalmethods are discussed and potential applicationswhere there is a possibility of errors due to thesemethods have been listed. A brief set of application ∗ Address all correspondence related to this author. areas which have not been explored through the lensof fluid structure interaction also have been discussedat the end of this article.Keywords: Multi-physics problems, Fluid StructureInteraction(FSI), Bolus transport, Discretization.
Nomenclature e Shear strain in fluidE Youngs Modulus of solidf External forcesG Bulk Modulus of solidn Normal direction to surfacep Hydrodynamic pressuret timev Velocityx co-ordinatessuperscriptsf fluid domains solid domainsubscriptsi,j directions in tensor formGreek Symbols δ Kronecker delta (cid:15) lateral strain inn solid λ Lame Constant µ Dynammic viscosity ν Poissons ratioΩ closed surface/volume a r X i v : . [ phy s i c s . c o m p - ph ] J un denisty τ Shear stressΓ Interface σ Hydrodynamic stress
Major research in the domain of Energy Harvest-ing, Transportation and Medical science involvesmulti-physics and interactions between a fluid anda solid. Modern research in disease diagnosis andtreatment has helped us to take a strong forwardleap in healthcare and this medical research owesa part to fluid structure interaction. Interactionbetween a fluid and a solid occur on day to dayexperiences like movement of leaves due to wind,movement of flag in air, production of sound etc.and in most complex engineering applications likeflutter of airfoil and energy harvesting.Fluid Structure Interaction (FSI) is a class ofmulti-physics problem where dependence betweenfluid and structural part plays an important role.The deformation and displacement of a struc-ture is affected by the forces applied by the fluidflowing past through it and change in geometryor position of solid inturn has an effect on fluidflow pattern. Thus, there is a two way couplingthat exists between fluid and solid and capturingthis through computational techniques can havehumongous applications in engineering design andmedical field. Various examples of FSI found inour nature include locomotion of aquatic animals,chirping and flying of birds, dispersal of pollenand seeds by wind, circulation of blood throughthe entire body, beating of heart etc. which hasinspired humans to mimic them and build efficientmachines. Engineering applications of FSI includedevelopment of energy harvesting devices, designingunderwater vehicles to decrease drag and improveperformance, design of turbine blades for variousapplications, design of high rise buildings to avoidvibrations etc. moreover in medical field, we findapplications in diagnosis of various cardiovasculardiseases (CVD) like atherosclerosis, arteriosclerosis, aneurysms, atherogenesis and prediction of futuregrowth prospects. In recent years, FSI has beenused to simulate sound production from vocal cordsand determine the physics behind bolus transportin esophagus. Also, there has been growing use ofFSI for surgical planning and treatment in varioussectors. The above mentioned applications havebeen discussed in detail in later sections of thisreport. Thus, it becomes very important to developpredictive FSI techniques that provides solutions tothe major concerns faced by modern era.For independent problems on fluid or solid, therehave been some analytical results [1] which helpus to understand the physics. In case of couplingbetween fluid and solid, there have been rarelyany analytical solutions except by Womersley [2].Another possibility of experimentation has beencarried out for some applications like vortex inducedvibration [3] but when it comes to medical appli-cations, invivo experimentation on humans seemshighly impossible. Thus, in such cases medicalimages obtained from computed tomography (CT)or magnetic resonance angiography(MRA) havebeen used to construct three dimensional geometryand perform computational experiments. From theabove argument we can infer that development ofcomputational techniques for FSI problems becomesvery crucial and taking the advantage of hugecontribution by scientific community towards thedevelopment of various numerical schemes whichcan efficiently handle the complexities involved ingeometry, material and fluid properties becomescrucial. The major challenge for FSI problems isthe coupling between structure and fluid besideensuring that boundary conditions are implementedaccurately at the fluid and solid interface. Therehave been different set of models developed to tacklethis issue and each one of them along with theirapplicability for certain kinds of problems has beenbriefed in the next section.igure 1: Computational Domain
Consider the whole domain of interest to be splitinto Solid (Ω s ) and fluid (Ω f ) with interface betweenthem(Γ) as shown in Fig. 1. The boundaries of fluidnot in contact with fluid are assumed to be rigid andthus a standard boundary condition may be appliedon them. The representation of the symbols pre-sented below are provided in the nomenclature. It is assumed that the fluid is in-compressible andpossesses a Newtonian rheology. The mass conser-vation associated with each cell leads to Continuitygiven by ∂ρv fi ∂x i = 0 (1)Momentum conservation at Control volume leadsto ρ Dv fi Dt = ∂σ fij ∂x j − f fi (2) Where Dv i Dt = ∂v i ∂t + c j ∂v i ∂x j σ ij = − pδ ij + τ ij τ ij = 2 µ ( e ij − δ ij e kk / e ij = ∂v j ∂x i + ∂v i ∂x j Consider the solid strcuture (Ω s ) to be elastic, ho-mogenous and isotropic. According to principle ofvirtual work , control volumes of solid have to satisfy ρ s Dv si Dt = ∂σ sij ∂x j − f si (3)Where σ sij = λδ ij ε ll + 2 Gε ll ε ij = 12 (cid:18) ∂u i ∂x j + ∂u j ∂x i (cid:19) G = E ν ) λ = Eν ν )(1 − ν ) There are mainly two conditions which have to besatisfied at the interface of any solid and fluid.1. Velocity of fluid boundary at interface must besame as the velocity of solid boundary(velocitydue to its displacement). This is generallytermed as kinematic boundary condition or noslip condition.2. The stress exerted by fluid at the interface shouldbe same as the external force which acts on solid.this is termed as Dynamic Boundary Conditionor stress continuity constraint.No Slip and stress continuity is applied at theinterface(Γ) as shown. si = v fi σ sij n i = σ fij n i There have been various computational methods tosolve equations governing the fluid dynamics [4] (usu-ally the Navier-Stokes equation) and structure dy-namics separately. They have been broadly calledas Computational Fluid Dynamics (CFD) and Com-putational Structure Dynamics (CSD). Coupling be-tween them started to pick up eventually when peo-ple started to think about the reason behind unfor-tunate events like the collapse of the Tacoma Nar-rows bridge, Vibration of deep water risers in oil rigsetc. and eventually computational techniques startedgetting developed. Major classification of numericalmethods is based on the way coupling between fluidand solid is introduced, the way grid is generated(i.e, whether it changes with fluid/solid domain) andthe way in which discretization has been performedto obtain the final algebraic equations to be solved.These classifications are discussed in detail below.
This classification is made on the basis of how themathematical framework has been designed to solvefor fluid and solid domains. In case of Monolithicapproach a unified mathematical governing equa-tions are defined for both fluid and solid and thewhole domain is solved as one entity. The mainadvantage behind using this method is that theinterfacial conditions are inherently present in theformulated mathematical framework and a singlediscretization scheme can be applied. It has alsobeen observed [5] that a better accuracy is observedin case of well defined multi-physics problems. Oneof the main issues with this approach is that thecode development becomes heavily problem-specificand a sense of generality is lost. Also, some kind ofexpertise is required in the domain specific area to Figure 2: Monolithic and Partitioned Approachdevelop such an algorithm. Fig. 2 shows the basicflow in these two approaches.In case of partitioned approach, solid and fluidgoverning equations are solved separately in theirrespective domains and coupling between them isintroduced by applying suitable interface conditions.Discretisation scheme and meshing can be completelydifferent for both the domains but a sense of matchmust exist at interface location. If meshes are not ina matching condition at the interface, a ghost patch[6] is usually introduced wherein communication istransmitted from solid to fluid or vice versa throughthis ghost patch. This ghost patch is developed insuch a way that it matches solid and fluid domainson their respective sides. This method was devel-oped in the beginning stages of computational FSIdevelopment and is still being applied and developedsince it has an inherent advantage of using robustfluid and solid solution algorithms which have beendeveloped over the past few decades. These methodsare prone to errors especially at the interfaces due tomismatch of discretisation and mesh. Another issuewith this method is that the computational costis high compared to that of Monolithic approachsince interface conditions needs to be applied everytime and remeshing of the whole domain has to bedone either for every time step or when there is alarge displacement which could possibly disturb theskewness and other properties of the mesh. .2 Conforming and Non-ConformingMesh
A classification based on the type of mesh gen-erated to solve fluid solid interactions leads us toconforming and non-conforming type mesh (Fig. 3).In case of conforming type system, the interfacebetween fluid and solid is considered to be a physicalboundary. In most of the cases, fluid particles aretracked through Eulerian approach whereas solidmovement is tracked via Lagrangian formulation.Boundary conditions are applied at the interfacefor both fluid and solid governing equations andthus the communication between both the domainshappens through these boundary conditions. Fluidand solid domains can utilise various mesh types butneeds to be updated at every/regular time-steps.For example, if a body contracts the mesh needs tobe updated to either remove the solid cells outsidecontracted body or to contract the existing mesh.This task becomes cumbersome when there are alarge number of elements in mesh and deformationis high. Tracking of the boundary is also essentialwhile going to the next time step. Methods underthis category include Arbitrary Lagrangian Eulerian(ALE)([7], [8]), Deformed Spatial Domain/Stabilisedspace time (DSD/SST) [9] etc. which will discussedin the upcoming topics.In case of non-conforming mesh methods, theinterface conditions are not treated as physicalboundary conditions for fluid and solid domain.They are treated in governing equations itself so asto give the same effect of boundary conditions (noslip and specified traction). The major advantagewith these methods is that though fluid and solid arediscretized quite differently, a single mesh is enoughto solve the problem of FSI. Thus, there won’t beany issue of re meshing in this type which wouldbe very effective in reducing computational cost.Efficient mathematical formulation of governingequation for accurate interface conditions needs tobe considered. Various methods have been devel-oped under this category which include ImmersedBoundary Method(IBM) [10], Immersed InterfaceMethod(IIM) [11], Coupled-Momentum method [12], Figure 3: Non-Conforming (top) and Conforming(Bottom) meshDirect Forcing method etc. which will be detailed inthe next topic.
After the development of some initial Fluid-Structure-Interaction techniques based on a naiveapproaches, various discretisation techniques startedgetting developed for specific problems to improve ac-curacy. Some of these methods which are applicablefor general FSI problems are discussed below.
In the initial stages, this method was developed toobtain solutions for interfacial and free surface flows.Hirt et al. [7] brought ALE method which was appli-cable to all flow regimes. It used a Finite Differencediscretisation in which nodal points could move withfluid or be held at fixed position or could be movedin a specific direction as specified by user. This ledto the generation of word ‘Arbitrary LagrangianEulerian’. Based on the value of mesh velocity,the discretization became either fully Eulerian orully Lagrangian or hybrid. The computationsin a particular time step have been performed inthree separate phases. These phases include explicitvelocity computation, implicit calculation of pressureand updating the velocity and rezoning using meshmovement. The method was tested on several caseswhich included shock tube flow with moderate andvery large sound speed and flow around a rectangularblock. Stability analysis was also performed withthis method.Kennedy et al. [8] developed a computationalmethod to analyse the effect of pressurised bubbleon a structure used in reactor vessels. They usedFinite Element discretization on the fluid domainand for energy equation. After obtaining fluid solu-tion, displacements of solid was calculated throughtraction force exerted by fluid. After this step, meshvelocity and co-ordinates were updated to performremeshing. The numerical method was tested ona fluid inside a rigid and flexible walls. A briefalgorithm followed by them is as follows:1. Calculate acceleration of fluid as ∂ v ∂t = M − (cid:0) f ext − f int (cid:1) (4) M is found through Navier Stokes equation(2.2).2. Obtain fluid velocity vv n +1 / = v n − / + ∆ t n ∂ v n ∂t (5)3. Set Grid Velocity v G as some fraction of fluidparticle velocity at interface obtained throughstructure solution.4. Update mesh point co-ordinates as x = x + ∆ t n +1 / v G,n +1 ∆ t n = 12 (cid:16) ∆ t n +1 / + ∆ t n − / (cid:17)
5. March to the next time step. Most commonly used discretization schemefor solids has been finite elements because of itsrobustness to handle material and geometric non-linearities. There have been some attempts [13] todiscretize solid through Finite Volumes (FVM) andcompare the accuracy with that of Finite ElementMethod(FEM). The main motivation behind usingsuch a discretization is to allow efficient and smoothcoupling at the interface of solid and fluid (whichis mostly discretized through Finite Volumes in anEulerian frame). Tsui et al. [14] adopted FiniteVolume discretization and partitioned approach forboth fluid and solid domains. Their method had acontrasting uniqueness where fluid variables wereplaced at grid centroids whereas solid variables wereat the nodes of cell. They successfully validatedresults for flow past a cylindrical bluff body withplate fixed on the rear side.Hubner et al.[5] developed a monolithic approachfor FSI problems where-in simultaneous solutionwas achieved for fluid and solid. Remeshing wasperformed before marching to the next time step.Discretization was performed through space timefinite elements [15]. The simultaneous solutionwas possible by formulatining fluid and solid dis-cretized equation into a single equation. The solidwas modelled with a complete Lagrangian fashionwhere displacement and Piola-Kirchhoff stresseswere the variables to be solved for. Fluid domainwas modelled in an Arbitrary-Lagrangian Eulerianrepresentation where boundaries of fluid domainintersecting with solid were in Lagrangian mode.The method was validated by solving flow over abuilding roof, and through vortex excited elasticplate.Though these methods are found to be accu-rate and easier(as they use existing solvers of fluidand solid domain), remeshing has to be carriedout at successive interval time which increasesthe computational cost by a huge margin. Thesemethods encounter issues for interface matchingdue to which significant accuracy at interfaces islost. When we deal with moving bodies (eg. objectfalling under gravity) or with large deformationroblems (expansion of air inside balloon etc.,),mesh generated at different intervals may be of lesserorder or lead to entanglement (when solid rotates).Thus, in these cases, ALE is found to be not verysuitable.
The IBM is a Non-Conforming methods that usesa Cartesian grid on whole domain and boundaryis approximated either as a curve(for 2D flow) ora surface(3D flow). Structure is solved separatelyto update the coordinates of curve. Further, thegoverning equations in IBM are modified so as toincorporate the boundary conditions of no-slip by in-troducing source terms. This method was introducedby Peskin [16] where blood flow patterns in a modelheart was analysed. The method used Eulerianframe for fluid (blood) that is discretized by finitedifferences and applied Chorin’s projection methodfor obtaining fluid solution. Solid was modelledin Lagrangian frame where forces generated weremodelled through combination of springs.Immersed Boundary Method is broadly classi-fied in two types based on the manner in whichforcing term is applied (refer Fig. 4). In the firstmethod, which is popularly known as ‘ContinuousForcing’, force term developed through solid at theinterface is applied on the fluid momentum equation(Eqn. (2)) and then a suitable discretization isperformed. The following equations may help tovisualise the flow of solution scheme with thisapproach—[ L ]( v i ) = ∂v i ∂t + v j ∂v i ∂x j + 1 ρ ∂p∂x i − µρ (cid:32) ∂ v i ∂x j (cid:33) (6)A forcing term is introduced as follows[ L ] { v i } = { f b } (7)The forcing terms are interpolated from solid domainusing dirac delta functions as follows (k referes tothe discrete point on interface curve). f m ( x , t ) = (cid:88) k F k ( t ) δ ( | x − X | ) (8)Where ‘x’ denotes co-ordinates in fluid do-main(eulerian) and ‘X’ denotes co-ordinates insolid field(Lagrangian). Various algebraic expres-sions are presented for ‘ δ ’ by Peskin [10]. Velocity atthe Lagrangian points is obtained through ∂ X k ∂t = v ( X k , t ) (9)It was observed by Mittal and Iaccarino [17] that con-tinuous forcing approach was suitable for immersedelastic bodies and moving boundaries immersed ina fluid but produced errors when simulating rigidbodies and bodies with complex physical boundariesas it softens the interface.Another class of IBM are the discrete forcingapproach based models where a normal fluid govern-ing expression without forcing terms are discretizedfirst and then a modified forcing term is applied todiscretized equations via dirac delta kernels (so thatforce applies only on boundary and near boundarynodes). This method further has classifications likeindirect boundary condition imposition where fluidvelocity is predicted without forcing terms and thencorrected through suitable usage of kernel functionsfor force and direct boundary condition impositionmethods which prevent smoothing of boundaries.This indirect boundary condition imposition isspecially used for boundary layer problems like flowthrough vented cylinders, pectoral fin locomotionanalysis. The broad classifications of IBM have beenprovided below. After the introduction of Immersed Boundarymethods, it was observed that Pressure and viscousstress were discontinuous at the interface and IBMregularized this discontinuity. The main reasonbehind this was the usage of discrete dirac deltafunctions for singular force term calculation at theinterfaces. Thus, it was found to have first-orderigure 4: Flowchart of Immersed Boundary Methodaccuracy at interfaces [11]. Though sharp interfaceIBM was developed [10], full second order accuracywas not possible. The initial Immersed InterfaceMethod (IIM) was developed by Li [18] but wasrestricted to simpler elliptic PDEs applied to phasechange problems and stokes equations. Li andLai [19] developed IIM for Navier-Stokes equationwhich was able to achieve second-order accuracyat interface. The major modification with respectto IBM was that the jump conditions of variablescalculated through Taylor series were applied atthe interface instead of discrete delta functions.The derivation for jump theorem and calculationof jump are reported in Li and LeVeque [20]. Themethod was successfully applied on moving interfaceproblems that involved phase changes.A recent study by Griffith and Patankar [21]showed that there were some discrepancies in thesolution obtained for flow past cylinders with IBM.They stated that this problem of leakage of mass fluxinside cylinder could be easily overcome by the usageof IIM. When a channel flow was tried to be solvedthrough IBM [22], it was observed that velocitydiffused outwards of channel walls. They introduceda novel formulation for solving Fluid-Structure Inter-action problems through Immersed Interface Methodby calculating jump conditions from projections.Performing simulations with discrete IB formulationreveals that results near the interface have lowerorder accuracy and thus produce erroneous results. In case of Vortex Induced Vibration problem, it wasobserved to have some internal flow which is notdesirable. In case of channel flow, deviation fromanalytical results were observed in flow rate with theuse of classical IB formulation. They have projectedthe jump conditions on the subspace containinginterface and then derived the jump conditions. Thisadjustment has been done to avoid the discontinuityin the interface normal directions. Ray-CastingAlgorithm has been used to find the intersectionbetween finite difference stencil and the interface.They have used IBAMR software to incorporatetheir new methodology and carry-out simulations.Many fundamental flow problems like flow in 2Dand 3D channel (straight and inclined), flow overstationary and rotating cylinder have been solvedwith the proposed as well as classical IB method.Better results (in terms of sharpening of boundaryand avoiding internal flow) were observed withIIM (with projected jump conditions applied on allvariables). It was observed that solution was glob-ally second order accurate for velocity and tractionvariables whereas the jump conditions with leastorder were considered using discretisation. This wasachieved due to the projection of jump conditionsacross the interface. Thekkethil and Sharma [23]developed Level-Set based IIM formulation to solveFSI problems involving flexible bodies. Boundaryconditions at interface were applied by the LevelSet(LS) function which was calculated based on thegeometry of solid.
Although there have been some Non-Conformingtechniques like IBM and IIM which can cut thecomputational cost and make programming clearer,coupling accuracy especially while using differentdiscretizations can be error prone at interfaces. Anew method which is monolithic in nature and avoidserrors at interfaces was developed by Figueroa et al.[12] and is popularly called as coupled momentummethod. Using the thin wall approximation, thelateral body of solid surface (Eqn. (10) ) is projectedonto the fluid surface using quantitative wall thick-ness. This process is approximated as Ω s ( . . . ) dx = ζ (cid:90) Γ s ( . . . ) ds (10)The Nominal governing equation for fluid is writ-ten in integral form of Finite Element discretisationapproach. Traction boundary condition t f is broughtinto the governing equations for fluid. It was assumedthat surface traction t f acting on the lateral surfaceis equal and opposite to vessel wall surface tractioni.e, t f = - t s . This inturn was incorporated in bodyforce term of solid governing equation and an expres-sion for t f was obtained. This force was substituted influid-governing equation. Thus, a common governingequation for fluid and solid was thus obtained. Sim-ulations were performed on idealized model of com-mon carotid artery and deformation model was com-pared with that of rigid model. Resistance, Impe-dence and Pressure type boundary conditions wereused and displacement of artery wall along with flowrates were measured. Recently a verification studyfor this method was published by Filonova et al. [24].Some challenges faced by this method include incor-porating spatially varying material properties and de-velopment of method for varying thickness models. In this section, an attempt has been made to coversome important applications involving interactionsbetween fluid and solid. The research gap which ex-ists in individual application domains have been triedto be addressed so that it would pave way for workingon those areas to provide solutions. The applicationsare presented below in separate subsections for VIVinduced energy harvesting, nature inspired vehiclesand bio-medical applications.
After the incident of Tacoma Narrows Bridge in1940, scientific community started to study thereason behind the collapse of reason. One theory suggested by community suggested the sheddingfrequency of swirls behind the bridge were actuallymatching the natural frequency of vibration of bridgebecause of which it started to resonate. The studyof vortices and the effect it has on a structure thusbegan which led to the development of study relatedto interaction between fluid and solid. Study ofVIV has been applied in various fields which includeprevention of vibrations of vertical structures likechimney stack, offshore risers and harvesting energy(called vortex power).Vortex induced vibrations were initially studiedanalytically by Facchinetti et al. [25] where theyconsidered a lower order equation for structuralvibration and wake vibration (van der Pol equa-tion). Authors considered three different models forcoupling wake and structure oscillations throughvelocity, displacement and inertia terms. Theanalytical results from these three models werecompared with experimental results for variousparameters. It was observed that inertia modelbased on acceleration coupling yielded the bestresults. A major disadvantage of these modelswas the ignorance of some terms during derivation(like non-linear variables) and approximation ofsome parameters based on experiments. Hence,a computational approach applicable to all set ofproblems becomes necessary.Herjfold et al. [26] developed a cost effectivecomputational methodology for analyzing the VIVphenomenon on deepwater risers placed in offshoreplants so as to test the efficiency of riser designbefore actually developing it. Authors consideredfinite number of two dimensional fluid planes alongthe length of riser on which incompressible Navier-Stokes equations were solved separately. Fractionalstep algorithm was used using NAVSIM program tosolve for velocity and pressure. The values of surfaceforce obtained on fluid simulation were interpolatedon surface of structure. Displacement field for solidwas computed based on the interpolated surfaceforces using algorithm of USFOS (Ultimate strengthof framed offshore structures). Validation of thedeveloped model was performed by comparingtructure response at various sections with the oneobtained from experiments. A small amount ofunder prediction was observed at bottom of theriser. This might have happened due to ignorance ofground effects.Zhu and Peng [27] developed a computationalmodel to analyse the effect of heaving motion offoil for the purpose of efficient vortex control. Themathematical formulation of Navier-Stokes equationwas performed in XY-coordinate system which wasfixed in space and a xy-coordinate system whichwas fixed with foil. Discretization was performedbased on finite difference. In order to incorporateinteraction between fluid and structure, an iterativecoupling model was used. This model followedsome steps wherein initial excitation was providedto the foil and which the deflection (heave) of foilwas computed based on the lift force acting on thesurface of foil at different sections. They came to aconclusion that partial recovery of energy of flappingcould be recovered when leading edge vortex is madeto be formed as far as possible from the pitchingaxis.Williamson and Govardhan [28] discussed the overalldevelopment in terms of computational and experi-mental methods to study the VIV phenomenon soas to apply the findings in industrial applications.Review of methods was performed based on theireffectiveness to satisfy the Griffin-plot and other pa-rameters. It has been observed that though sufficientcomputational techniques have been developed toaddress VIV, much efforts on high Reynolds numberVIV with inclusion of turbulence model throughDirect Numerical Solution or other methods is theneed of the hour to appreciate the real world flows.This might lead us to efficiently develop Industrialproducts with more confidence.
Requirements for Enhancing the lift and reducingdrag in mechanical systems for locomotion has ledthe scientific community to find inspiration fromaquatic and flying species on Earth. An aerodynamic model was proposed by Weis-Fogh [29] through whichsmall insects like wasps were found to enhance liftduring their flight. He proposed that insects usedclap and fling motion during their flight to optimiseperformance. Miller and Peskin [30] used a ‘targetboundary’ version of Immersed Boundary Methodto simulate clap and fling in a tiny wasp (calledEncarsaria formosa). They performed simulationat various flow conditions (8¡=Re¡=128) to analysethe lift enhancement mechanism. Displacement androtation of the wings were specified to understandthe interaction between fluid and wing structure.Lift and Drag coefficients were obtained as theoutputs which were validated with experimentalresults. It was found that leading and trailing edgevortices existed for single wing case throughoutthe cycle whereas only leading edge vortex (LEV)was present in two wing case during initial flingmotion. Vortical asymmetry during translation anddominance of leading edge vortex were found to bethe reason for lift enhancement. Results obtained forone wing and two wings at various reynolds numbershowed that clap and fling mechanism was mostsuitable for insects flying at lower speeds.Efficient propulsion found in aquatic animals isattributed to the optimal vortex control done bythem through movement of body and tail fin. Inorder to understand the mechanism behind effectivepropulsion system of fish and derive the optimalparameters for body motion, Triantafyllou [31]performed a systematic review of experimentalobservations and computational methods. It wasshown that when an oncoming vortex moves overa foil (which has generated its own vortex), it caneither interact constructively to form a large trailingedge vortex or interact destructively to weaken thevortex or form pairs to broaden the wake region. Theexperiments carried out on oscillating and thrustproducing foils showed that there was an optimumSt (around 0.25 - 0.35) where maximum propulsiveefficiency was noted.Bhalla et al. [32] developed a computationalFluid-Structure Interaction model using cartesianbased Immersed Boundary Method and applieddaptive Mesh Refinement to capture thin Bound-ary Layer at Fluid surface Interface. They validatedtheir model by applying it on a 3D model of blackghost knifefish. The fluid was modelled in an Eule-rian framework whereas solid was modelled throughLagrangian framework. Specified motion is providedto the body and force term fc is incorporated ingoverning equations to account for that whereas aterm fb is used to account for deformations of solidbody. Reverse karman vortex street is observed ineel locomotion close to trailing edge which suggeststhe self propelling nature observed in eels. Forwardand Backward motion of black ghost knifefish wassimulated to understand its ability to swim in thereverse direction.Recently Thekkethil et al.[33] performed a uni-fied hydrodynamics study based on various typesof motions (pitch, heave and undulation) throughan in-house code based on Level Set Immersedboundary formulation. Apart from solving regularNavier stokes equation, two separate equationsbased on advection velocity at the interface andlevel-set function were used to classify the domains.A non dimensional number based on wavelength ofundulation was used for unifying various motions.Simulations were performed at several St andpropulsive efficiency was derived. The role of reverseKarman street in enhancement of propulsion wasdescribed by authors. A sensing mechanism used bypredator fish was also proposed after visualising thevortex in the downstream direction. Performancecharacteristics of self propulsive motion of NACAairfoil (which represents fish cross-section) has beeninvestigated by Thekkethil et al. [34] using thein-house developed Level Set (LS) based ImmersedInterface Method. It was observed that pitchingtype fishes have higher propulsive efficiency andstability during the initial phases of propulsion.They extended the study to 3D batoid type fishmotion [35] with an added parameter of 3D aspectratio. Horshoe type vortices with multiple vortexrings were observed for pitching type batoids whereasthe propulsive efficiency was found to be high forintermediate wavelength numbers and larger aspectratios.
Cardiovascular ailments have become a major reasonfor deaths in the modern world mainly due to thesedentary lifestyle and food habits. It was estimatedby World Health Organisation (WHO) that around7.3 million deaths are reported annually aroundthe globe due to coronary artery abnormalities [36].Many numerical and experimental works have shownthere exists a strong correlation between Hemo-dynamic parameters like Wall Shear Stress(WSS),Oscillatory Shear Index(OSI) , Relative ResidenceTime (RRT) etc. on the growth and progress ofdiseases. Cardiac-Surgeons are usually referred to asCardio-Vascular Fluid mechanicians who brings theblood flow to its normal condition either by insertinga mechanical device which enlarges artery(calledstent) or by making some other way for bloodflow(Grafting). Since in-vivo measurements are dan-gerous and very difficult, it becomes utmost builda most accurate computational model which cansimulate blood flow through arteries so as to predictpotential sites for disease progression, help surgeonsin surgery planning and study the causes for diseasedevelopment. Major challenges for computationaldevelopment include using patient specific models,using proper rheology of blood, modelling bloodconstituents and considering proper compliance ofarteries through accurate artery material properties.Inclusion of compliance leads us to requirement ofcoupling fluid and structure interaction. Crosettoet al.[37] used ALE (Section 3.1) to calculate thevariation of mean velocity and pressure at differentsection with respect to the cardiac cycle. Patientspecific arteries were modelled using ITK Snapsoftware. Simulations were performed through LifeVmulti-physics software. A hybrid Boundary condi-tion based on pressure and traction was formulationto obtain realistic conditions. Blood was assumed tohave a Newtonian rheology as the arteries consideredwere of sufficiently larger cross section. It was ob-served that when the inlet pressure was specified asboundary condition, there was a diastolic backwardflow which is not usually observed in fixed wallsimulation. This phenomenon was actually reportedn human blood circulation cycle which led to closingof aortic valve. It was also shown that WSS wasoverestimated by rigid wall model. Thus, it can beargued that rigid wall models underestimates thedisease growth (as CAD is associated with low andoscillating shear stress zone).Bathe and Kamm [38] Studied the effects ofaortic stenosis severity, asymmetric nature and inputpressure conditions on the compressive and tensilestresses, blood flow patterns and plaque rupturepossibilities. Computational method developed onFinite element and finite difference scheme wasvalidated with experimental results. Poly VinylAlcohol(PVA) - hydrogel was selected as the arterymaterial for experiments. Pressure measurementswere obtained through manometric techniqueswhereas flow rates were measured through ultra-sound flow meter. Three dimensional Mooney-Rivlinmodel was used to model arteries computationally.Fluid domain was solved through finite differencescheme in a curvilinear coordinates. FSI at interfaceswas implemented through incremental boundaryiteration. Maximum compressive stress was observednear the inner wall at stenosis throat for asymmetricartery whereas point of maximum compression wasobserved to be in the near post-stenotic region. Tur-bulence and Non-Newtonian rheology were ignoredwhich have effects on hemodynamic parameters assuggested by Mahalingam et al. [39].Based on some of the fine literature discussedabove, it can be observed that there have been min-imal efforts to consider all parameters into accountto simulate realistic blood flow in compliant arteries.A major facet ignored in currently available studiesinclude the effect of shear induced migration of redblood particles [40] in complaint stenotic arteries inthe carotid region.
Phonation refers to the production of sound dueto fluid induced vibration of the vocal cords inthe larynx. Structure of human larynx consists ofthree different layers [18] as shown in figure. Vocal cord paresis occurs due to damage on the upperepithelium and leads to disability in speaking. Thisproblem is usually found during injury or due to oldage. In the process of phonation, vocal cord is madeto vibrate by the alternate vortex shedding in thewake of the vocal cords. Medialization laryngoplastyis a surgical treatment for vocal cord paresis wheresurgeon tries to insert an implant made of a smallstrip which can make up of the damaged mass ofepithelium thus enabling normal sound production.This process is iterative in nature and requires ahuge experience of surgeon to handle it. Simulationof vocal cords with particular implant can help thesurgeon to plan the surgery and remove the iterativeprocedure.In the initial stages, fully coupled FSI approach wasused by Ishizaka and Flanagan [41] where airflowwas considered in one dimensional and two lumpedmasses for vocal cords were used. Mittal et al. [17]developed a complete two-dimensional FSI modelsharp interface immersed boundary formulation.Incompressible NS equations for air discretizedthrough sharp interface immersed boundary methodwas coupled with cartesian BM formulation per-formed on structure. Appropriate properties basedon experiments were used to model vocal folds.Triangular elements were used to discretize solid andsecond order finite difference scheme was applied fortemporal and spatial discretisation. An equationfor displacement eigen mode was formulated andfrequency was calculated. Normal modes foundthrough experiments were compared with the eigenmodes obtained through simulation. It was observedthat second and third eigen modes obtained in sim-ulation were in reversed order from normal modes.Thus, a modification in numerical method to handlesolid is needed for accurately solving this issue.
Esophagus is a thick multi-layered cylindrical tubewhich helps to transfer food (called bolus in medicalterms) from pharynx to stomach through pulsatilewall motions controlled by neural stimulus. Esoph-gus is composed of mucosal, Longitudinal muscle(LM) and Circular muscle (CM) layers which shortenand contract simultaneously to produce the desiredwall motion [18]. It was found that around 5-8percentage of the population above 50 years of agehave swallowing difficulty which is associated withesophageal dysphagia [42]. Understanding the roleof muscle contraction and shortening in esophagealtransport can be achieved through numerical sim-ulations. Quantification of disease pathologies andplanning treatment strategies are the next steps inthis technology which have to be explored.Kou et al. [43] developed a computational FSImodel to simulate peristaltic wall motions in esoph-agus using Immersed Boundary approach based oncontinuous formulation (refer 3.3.2). In this model,bolus was considered as a viscous fluid and esophagusas a thick tube containing fibers at different orien-tation. This whole setup of fluid inside a solid wasthen submerged inside another fluid where equationswere not solved. Numerical method was validatedwith analytical solution for tube dilation problem.The Immersed Boundary method developed containsmomentum and incompressibility of coupled fluidand structure in an Eulerian framework whereasstructural forces are modelled through Lagrangianframework.The three different fibers of esophagusare modelled as springs and beams whose whosestiffness parameter was derived through materialproperties (Young’s modulus and poisson’s ratio).Finite element formulation of wall structure usingtriangular elements has been performed based on theshape function calculated through local coordinates.Muscle activation kinematics is used for wall motionthrough which displacement is specified at the innerlayers of Esophagus. Numerical challenges due to awide range of dimensional quantities in wall diameterand esophagus length were observed due to whichsome amount of fluid leakage was observed. Anadaptive mesh technique may prevent this issue andkeep computational cost in check. Kou et al.[44]validated their numerical pressure contours withclinical manometry measurements.
A brief overview of different numerical methodsused to solve the multi-physics problems of fluid andstructure have been discussed with some importantapplication domains where computational methodsbecome very essential. The scope and limitationsof the various methods classified based on couplingmanner and discretization have been discussedin detail. Methods suitable for each applicationdomain and issues which persist even today withthese models have been listed. A list of standardnumerical methods, their developers and majorapplication domains which use these methods hasbeen provided in Table 1 .
There has been very significant progress in the devel-opment of computational techniques and applicationof the developed methods to design useful mechanicaldevices which can harvest energy, decrease the vibra-tion of deepwater risers, build better infrastructuresand help us to understand the way birds, insects andaquatic animals locomote. Recently, there has beena huge advancement in the development of diagnosistechniques, surgical planning in medical domainwhich have some contribution from Fluid StructureInteraction based computational models. Here wepresent scope for future work on computational FSImethods and applications in separate subsectionsbelow.
1. Recently Tsui et al. (2013) have used FiniteVolume discretisation for solids in FSI problemand validated the numerical method with flowpast a circular cylinder with flexible plateattached on its lee side. These methods haveable 1: Summary of Literature for various numerical methods and applicationsNumericalMethod Significant Developers Application DomainsALE Hirt et al.[7],Kennedy et al.[8] Free surface flows, Sloshing dynamics, vibration of risers,Flapping wing aerodynamics.DSD/SST Tezduyar et al. [9] Modelling patient specific arterial blood flow, Flow throughWindsock.IBM Peskin [16] Blood flow through heart valves, Vortex induced vibration,Phonation and Esophagal transport.IIM Li and Lai [19] Hydrodynamics of aquatic locomotion, Intra Vena Cava (IVC)flow, Solidification and crystallization study.CMM Figueroa et al. [12] Deformable artery flow.an inherent advantage of obtaining betteraccuracy at interface since solid and fluid hasa common coupling scheme at the interface.These methods are not explored completelyin the domain of FSI and especially in thedomain of cardio-vascular FSI. Using Eulerianframework for solids and discretizing throughFVM remains an unanswered question whichcan have huge implications in accuracy andcomputational costs.2. It has been observed in most of the literaturethat FSI model are used to solve problemswhere fluid motion is assumed to be laminar.Hence, there are no relevant computationalmodel found which use turbulence model forfluid and then applying two way couplingwith solid. If we consider the case of pulsatileblood flow in small arteries like those found incerebrum and in deep veins, a small constrictionmay cause the flow to become turbulent. Thusturbulence modelling for these kind of biologicalFSI problems becomes necessary.3. It has been observed in Immersed Boundarymethod that though second order accuracy isobtained for smooth interfaces, calculations on sharp and especially discontinuous interfaces(usually found on walls with protrusions) cannotachieve this accuracy. Though there are someother methods like Immersed interface method,blob-projection method which can avoid this is-sue, a core Immersed Boundary method whichcan obtain translation invariance and second or-der accuracy needs to be developed.
1. Phonoangiography is a non-invasive diagnosistechnique in which arterial murmurs (calledbruits) are analysed to detect abnormalities.Though there was substantial research forqualitative phonoangiography [45], quantitativemethods which help to localise the source andquantify the abnormalities have been developedin the recent past through computationaltechniques [46]. A major assumption made bythem was ignorance of fluid induced vibration ofstructure which can lead to significant change insound spectrum. They had considered coronaryarteries which are well protected by the rib cagewhich may substantially reduce the resonancepeak created by structure vibration. If we con-sider the case of carotid arteries, this assumptionmight not hold true as it is located very closeo the epidermal surface where measurementsare made. Thus, analysis of bruits on carotidarteries with Fluid Structure model coupledwith acoustics can take us close to device aGuided User Interface(GUI) which can helpmedical practitioners to quantify abnormalitiesquantitatively.2. Considering Shear Induced Migration (SIM) ofRed Blood Cells (RBC) in small complaint ar-teries and calculating Hemodynamic parametersto predict growth of diseases can help surgeonsin medical planning. This problems requiresone to couple FSI model for blood-artery withthe mixture model which governs transport ofRBC to obtain final solution. This becomesvery important in small arteries where SIM isenhanced and may change the platelet activa-tion mechanisms which are the main reasons forplaque deposits.3. Development of new energy harvesting deviceslike tapered vented cylinders as vortex gener-ators need a very efficient FSI model whichcan handle geometric complexities. It wasproposed by Kumar et al. [47] that usage ofsuch systems can improve energy harvestingquantitatively but FSI model was not incor-porated by authors due to geometric complexity.4. Performance of new underwater vehicles op-erating on the strategy of aquatic locomotionneeds to be verified before actual design andmanufacturing. These include the use of flexibleplate on the lee side of cylinder as a fin forthese vehicles in order to reduce drag. Thoughsimulation has been performed on part scales, afull fledged simulation using FSI models is veryessential for moving ahead with next procedures.
References [1] Philip G Drazin and Norman Riley.
The Navier-Stokes equations: a classification of flows andexact solutions . Number 334. Cambridge Uni-versity Press, 2006.[2] John R Womersley. Method for the calculationof velocity, rate of flow and viscous drag in ar-teries when the pressure gradient is known.
TheJournal of physiology , 127(3):553–563, 1955.[3] CC Feng.
The measurement of vortex inducedeffects in flow past stationary and oscillating cir-cular and D-section cylinders . PhD thesis, Uni-versity of British Columbia, 1968.[4] Atul Sharma.
Introduction to computationalfluid dynamics: development, application andanalysis . John Wiley & Sons, 2016.[5] Bj¨orn H¨ubner, Elmar Walhorn, and Dieter Din-kler. A monolithic approach to fluid–structureinteraction using space–time finite elements.
Computer methods in applied mechanics and en-gineering , 193(23-26):2087–2104, 2004.[6] Gene Hou and Arunkumar Satyanarayana. Ana-lytical sensitivity analysis of a static aeroelasticwing. In , page 4824, 2000.[7] Cyrill W Hirt, Anthony A Amsden, andJL Cook. An arbitrary lagrangian-eulerian com-puting method for all flow speeds.
Journal ofcomputational physics , 14(3):227–253, 1974.[8] JM Kennedy and TB Belytschko. Theory andapplication of a finite element method for arbi-trary lagrangian-eulerian fluids and structures.
Nuclear engineering and design , 68(2):129–146,1982.[9] Tayfun E Tezduyar, Mittal Behr, S Mittal, andJ Liou. A new strategy for finite element com-putations involving moving boundaries and in-terfaces—the deforming-spatial-domain/space-time procedure: Ii. computation of free-surfaceflows, two-liquid flows, and flows with driftingylinders.
Computer methods in applied mechan-ics and engineering , 94(3):353–371, 1992.[10] Charles S Peskin. The immersed boundarymethod.
Acta numerica , 11:479–517, 2002.[11] Zhilin Li and Ming-Chih Lai. The immersed in-terface method for the navier–stokes equationswith singular forces.
Journal of ComputationalPhysics , 171(2):822–842, 2001.[12] C Alberto Figueroa, Irene E Vignon-Clementel,Kenneth E Jansen, Thomas JR Hughes, andCharles A Taylor. A coupled momentum methodfor modeling blood flow in three-dimensional de-formable arteries.
Computer methods in appliedmechanics and engineering , 195(41-43):5685–5706, 2006.[13] NA Fallah, C Bailey, M Cross, and GA Taylor.Comparison of finite element and finite volumemethods application in geometrically nonlinearstress analysis.
Applied Mathematical Modelling ,24(7):439–455, 2000.[14] Yeng-Yung Tsui, Yi-Cheng Huang, Chun-LungHuang, and Shi-Wen Lin. A finite-volume-basedapproach for dynamic fluid-structure interac-tion.
Numerical Heat Transfer, Part B: Fun-damentals , 64(4):326–349, 2013.[15] Thomas JR Hughes and Gregory M Hulbert.Space-time finite element methods for elastody-namics: formulations and error estimates.
Com-puter methods in applied mechanics and engi-neering , 66(3):339–363, 1988.[16] Charles S Peskin. Numerical analysis of bloodflow in the heart.
Journal of computationalphysics , 25(3):220–252, 1977.[17] Rajat Mittal and Gianluca Iaccarino. Immersedboundary methods.
Annu. Rev. Fluid Mech. ,37:239–261, 2005.[18] PHILIPPE Pouderoux, SHEZHANG Lin, andPETER J Kahrilas. Timing, propagation,coordination, and effect of esophageal short-ening during peristalsis.
Gastroenterology ,112(4):1147–1154, 1997. [19] Ming-Chih Lai and Charles S Peskin. An im-mersed boundary method with formal second-order accuracy and reduced numerical viscosity.
Journal of computational Physics , 160(2):705–719, 2000.[20] Randall J LeVeque and Zhilin Li. Immersedinterface methods for stokes flow with elasticboundaries or surface tension.
SIAM Journalon Scientific Computing , 18(3):709–735, 1997.[21] Boyce E Griffith and Neelesh A Patankar. Im-mersed methods for fluid–structure interaction.
Annual Review of Fluid Mechanics , 52, 2020.[22] Ebrahim M Kolahdouz, Amneet Pal SinghBhalla, Brent A Craven, and Boyce E Grif-fith. An immersed interface method for discretesurfaces.
Journal of Computational Physics ,400:108854, 2020.[23] Namshad Thekkethil and Atul Sharma. Level setfunction–based immersed interface method andbenchmark solutions for fluid flexible-structureinteraction.
International Journal for NumericalMethods in Fluids , 91(3):134–157, 2019.[24] Vasilina Filonova, Christopher J Arthurs,Irene E Vignon-Clementel, and C AlbertoFigueroa. Verification of the coupled-momentummethod with womersley’s deformable wall an-alytical solution.
International Journal forNumerical Methods in Biomedical Engineering ,2019.[25] Matteo Luca Facchinetti, Emmanuel De Langre,and Francis Biolley. Coupling of structure andwake oscillators in vortex-induced vibrations.
Journal of Fluids and structures , 19(2):123–140,2004.[26] Kjell Herfjord, SO Drange, and Trond Kvams-dal. Assessment of vortex-induced vibrations ondeepwater risers by considering fluid-structureinteraction. 1999.[27] Qiang Zhu and Zhangli Peng. Mode couplingand flow energy harvesting by a flapping foil.
Physics of Fluids , 21(3):033601, 2009.28] CHK Williamson and R Govardhan. Vortex-induced vibrations.
Annu. Rev. Fluid Mech. ,36:413–455, 2004.[29] Torkel Weis-Fogh. Quick estimates of flight fit-ness in hovering animals, including novel mecha-nisms for lift production.
Journal of experimen-tal Biology , 59(1):169–230, 1973.[30] Laura A Miller and Charles S Peskin. A com-putational fluid dynamics ofclap and fling’in thesmallest insects.
Journal of Experimental Biol-ogy , 208(2):195–212, 2005.[31] Michael S Triantafyllou, GS Triantafyllou, andDKP Yue. Hydrodynamics of fishlike swimming.
Annual review of fluid mechanics , 32(1):33–53,2000.[32] Amneet Pal Singh Bhalla, Rahul Bale, Boyce EGriffith, and Neelesh A Patankar. A uni-fied mathematical framework and an adaptivenumerical method for fluid–structure interac-tion with rigid, deforming, and elastic bodies.
Journal of Computational Physics , 250:446–476,2013.[33] Namshad Thekkethil, Atul Sharma, and AmitAgrawal. Unified hydrodynamics study for vari-ous types of fishes-like undulating rigid hydro-foil in a free stream flow.
Physics of Fluids ,30(7):077107, 2018.[34] Namshad Thekkethil, Atul Sharma, and AmitAgrawal. Self-propulsion of fishes-like undulat-ing hydrofoil: A unified kinematics based un-steady hydrodynamics study.
Journal of Fluidsand Structures , 93:102875, 2020.[35] Namshad Thekkethil, Atul Sharma, and AmitAgrawal. Three-dimensional biological hydro-dynamics study on various types of batoidfishlike locomotion.
Physical Review Fluids ,5(2):023101, 2020.[36] World Health Organization.
The world healthreport 2002: reducing risks, promoting healthylife . World Health Organization, 2002. [37] Paolo Crosetto, Philippe Reymond, Simone De-paris, Dimitrios Kontaxakis, Nikolaos Stergiop-ulos, and Alfio Quarteroni. Fluid–structure in-teraction simulation of aortic blood flow.
Com-puters & Fluids , 43(1):46–57, 2011.[38] Mark Bathe and RD Kamm. A fluid-structureinteraction finite element analysis of pulsatileblood flow through a compliant stenotic artery.1999.[39] Arun Mahalingam, Udhav Ulhas Gawandalkar,Girish Kini, Abdulrajak Buradi, Tadashi Araki,Nobutaka Ikeda, Andrew Nicolaides, John RLaird, Luca Saba, and Jasjit S Suri. Numer-ical analysis of the effect of turbulence transi-tion on the hemodynamic parameters in humancoronary arteries.
Cardiovascular diagnosis andtherapy , 6(3):208, 2016.[40] Abdulrajak Buradi, Sumant Morab, and ArunMahalingam. Effect of stenosis severity on shear-induced diffusion of red blood cells in coronaryarteries.
Journal of Mechanics in Medicine andBiology , 19(05):1950034, 2019.[41] Kenzo Ishizaka and James L Flanagan. Synthe-sis of voiced sounds from a two-mass model ofthe vocal cords.
Bell system technical journal ,51(6):1233–1268, 1972.[42] Ian J Cook. Diagnostic evaluation of dysphagia.
Nature Reviews Gastroenterology & Hepatology ,5(7):393, 2008.[43] Wenjun Kou, Amneet Pal Singh Bhalla, Boyce EGriffith, John E Pandolfino, Peter J Kahrilas,and Neelesh A Patankar. A fully resolved activemusculo-mechanical model for esophageal trans-port.
Journal of computational physics , 298:446–465, 2015.[44] Wenjun Kou, John E Pandolfino, Peter J Kahri-las, and Neelesh A Patankar. Simulation studiesof the role of esophageal mucosa in bolus trans-port.
Biomechanics and modeling in mechanobi-ology , 16(3):1001–1009, 2017.45] AO Borisyuk. Experimental study of noise pro-duced by steady flow through a simulated vas-cular stenosis.
Journal of sound and vibration ,256(3):475–498, 2002.[46] Jung Hee Seo and Rajat Mittal. A coupledflow-acoustic computational study of bruits froma modeled stenosed artery.
Medical & biologi-cal engineering & computing , 50(10):1025–1035,2012.[47] Kishan Ramesh Kumar, Sumant Morab, SuhasShekar, and Arun Mahalingam. Energy har-vesting from vortex induced vibrations usingvented cylinders mounted on light rail locomo-tive. In2016 7th International Conference onIntelligent Systems, Modelling and Simulation(ISMS)