Fundamentals of Fluid Fundamentals of Fluid Structure InteractionsStructure Interactions

Raghav. H. VRaghav. H. VFinal Year B. Tech Mechanical EngineeringFinal Year B. Tech Mechanical Engineering

Indian Institute of Technology MadrasIndian Institute of Technology Madras

Tutor: Prof. S. Tutor: Prof. S. MittalMittal

ContentsContents

Fundamental results and discoveries Fundamental results and discoveries concerning VIVconcerning VIVModes of vibrationModes of vibrationThe importance of mass and dampingThe importance of mass and dampingThe concept of a critical massThe concept of a critical massThe concept of The concept of ““effective elasticityeffective elasticity””

Fluid Structure InteractionsFluid Structure Interactions

E.g. Interactions between pipe movement and E.g. Interactions between pipe movement and the the unsteadyunsteady flow of a fluid within the pipe (water flow of a fluid within the pipe (water hammering, hammering, corioliscoriolis force meter, aircrafts etc.)force meter, aircrafts etc.)Initial cause of the interactions may be a Initial cause of the interactions may be a disturbance to the flow (e.g. valve or pump disturbance to the flow (e.g. valve or pump activity) or a disturbance to the pipe (e.g. activity) or a disturbance to the pipe (e.g. vibrating machinery) vibrating machinery) In both cases, the resulting dynamics will involve In both cases, the resulting dynamics will involve an extensive collection of pressure waves in the an extensive collection of pressure waves in the fluid and stress waves in the pipe wall fluid and stress waves in the pipe wall Interactions are inherently twoInteractions are inherently two--way way

Vortex Induced Vortex Induced VibrationsVibrations

What are Vortex Induced Vibrations?What are Vortex Induced Vibrations?What are the effects of VIV?What are the effects of VIV?What are the different kinds of VIV?What are the different kinds of VIV?How to eliminate VIV?How to eliminate VIV?

Vortex Induced VibrationsVortex Induced VibrationsUsually involves an Usually involves an interaction that is interaction that is dominantly onedominantly one--way way When fluid flows across a When fluid flows across a cylinder, vortices develop cylinder, vortices develop which are shed which are shed alternately from either alternately from either side of the cylinder side of the cylinder The resultant lift and drag The resultant lift and drag forces (coefficients forces (coefficients fluctuate) excite forced fluctuate) excite forced oscillations of the cylinder oscillations of the cylinder known as known as vortexvortex--induced induced vibrationsvibrations

A complex interaction A complex interaction --dependent on both flow dependent on both flow parameters and parameters and mechanical properties of mechanical properties of the cylinder the cylinder When the vortexWhen the vortex--induced induced vibration frequency nears vibration frequency nears one of the natural one of the natural frequencies (oscillating frequencies (oscillating frequency) of the frequency) of the structure, resonance structure, resonance happens happens –– also known as also known as locklock--inin, enhancing the , enhancing the vibration amplitudevibration amplitude

OffOff--shore oil and gas drilling and production face shore oil and gas drilling and production face hazards caused by exposure of submerged rig hazards caused by exposure of submerged rig components to underwater currentscomponents to underwater currentsCritical among these components are the Critical among these components are the

marine risersmarine risers, consisting of a series of steel , consisting of a series of steel pipes linking the surface platform to the sea bed pipes linking the surface platform to the sea bed During drilling, a marine riser contains the drillDuring drilling, a marine riser contains the drill--string and carries the string and carries the mudmud and debris from the and debris from the rock face, while during production it carries the rock face, while during production it carries the oil or gasoil or gasThe integrity of the marine riser under a variety The integrity of the marine riser under a variety of conditions is crucial to the entire operationof conditions is crucial to the entire operation

Is of practical interest to many fields of Is of practical interest to many fields of engineering engineering Can cause vibrations in heat exchanger tubesCan cause vibrations in heat exchanger tubesInfluences the dynamics of riser tubes bringing Influences the dynamics of riser tubes bringing oil from the seabed to the surfaceoil from the seabed to the surfaceIs important to the design of civil engineering Is important to the design of civil engineering structures such as bridges and chimney stacksstructures such as bridges and chimney stacksCan cause largeCan cause large--amplitude vibrations of amplitude vibrations of tethered structures in the ocean tethered structures in the ocean

Vortex Flow MeterVortex Flow Meter

A small bluff body of known dimensions is kept A small bluff body of known dimensions is kept in the flowin the flowVortices now get shed from the body, and the Vortices now get shed from the body, and the frequency of shedding depends on the velocity frequency of shedding depends on the velocity of flow and shape of the bodyof flow and shape of the bodyA microphone kept behind the body picks up A microphone kept behind the body picks up pressure fluctuations due to the vortices formedpressure fluctuations due to the vortices formedFrom the frequency of vibration, the flow velocity From the frequency of vibration, the flow velocity is estimatedis estimated

Solving the FSI ProblemSolving the FSI Problem

Principal Modes of Principal Modes of VIVVIV

2S 2S -- 2 single vortices formed per cycle2 single vortices formed per cycle2P 2P -- 2 vortex pairs are formed per cycle 2 vortex pairs are formed per cycle P+S P+S –– One single vortex and one vortex One single vortex and one vortex pairpairEquation of motion to represent VIV of a Equation of motion to represent VIV of a cylinder oscillating in the transverse Y cylinder oscillating in the transverse Y directiondirection

my cy ky F+ + =&& &

In the regime where the body motion is In the regime where the body motion is synchronized with the periodic fluid force, a synchronized with the periodic fluid force, a good approximate solution for the equation isgood approximate solution for the equation is

F(tF(t) = F) = F00 sin (sin (ωωtt))y(ty(t) = y) = y00 sin (sin (ωωtt -- φφ))φφ is the phase lagis the phase lag

The amplitude of the vibration, in terms of nonThe amplitude of the vibration, in terms of non--dimensional parameters is given bydimensional parameters is given by

where where

CCAA is the added mass coefficientis the added mass coefficientCCEA EA is the effective added mass coefficientis the effective added mass coefficient-- It includes the apparent fluid force in It includes the apparent fluid force in phase with the body accelerationphase with the body acceleration

Time Domain Time Domain ApproachApproach

In the time domain approach, we solve the system In the time domain approach, we solve the system equation in the time domain equation in the time domain A typical outcome of a timeA typical outcome of a time--domain analysis may be domain analysis may be a series of graphs showing how parameters vary a series of graphs showing how parameters vary with timewith timeThis is an intuitively obvious way to model real This is an intuitively obvious way to model real phenomena phenomena It is often an appropriate way too but not always the It is often an appropriate way too but not always the bestbest

Frequency Domain ApproachFrequency Domain ApproachHere, we solve the system equation in the frequency Here, we solve the system equation in the frequency domain, using the Fourier or the Laplace Transformsdomain, using the Fourier or the Laplace TransformsA typical outcome of a frequencyA typical outcome of a frequency--domain analysis domain analysis might be a series of graphs highlighting the dominant might be a series of graphs highlighting the dominant frequencies in the response of various parametersfrequencies in the response of various parametersIs a valuable representation because most responses Is a valuable representation because most responses are composed primarily of a small number of are composed primarily of a small number of dominant frequenciesdominant frequenciesSuch analyses can tell us very easily whether the Such analyses can tell us very easily whether the structure is likely to respond in a lively manner to structure is likely to respond in a lively manner to expected stimuliexpected stimuliThe time domain solution can be got by taking the The time domain solution can be got by taking the inverse transformsinverse transforms

Free vibrations of a cylinderFree vibrations of a cylinderThe amplitude of the vibration The amplitude of the vibration is related to the product of is related to the product of mass and dampingmass and damping2S and 2P modes are formed2S and 2P modes are formed2S with initial branch of 2S with initial branch of response and 2P with the response and 2P with the lower branchlower branchThere is a hysteretic transition There is a hysteretic transition between the two branchesbetween the two branchesLockLock--in happens when the in happens when the vortex shedding frequency is vortex shedding frequency is close to the natural frequency close to the natural frequency of the structureof the structure

Free vibration at low mass Free vibration at low mass and damping is associated and damping is associated with the existence of an with the existence of an upper branch of high upper branch of high amplitude response amplitude response appearing between the initial appearing between the initial and lower branchesand lower branchesThe frequency of the lower The frequency of the lower branch is not close to the branch is not close to the natural frequency and is natural frequency and is remarkably constantremarkably constantThere are three principal There are three principal branches and corresponding branches and corresponding two jump phenomenatwo jump phenomena

Critical MassCritical MassAs the structural mass As the structural mass decreases, the regime of decreases, the regime of velocity over which there velocity over which there are largeare large--amplitude amplitude vibrations increasesvibrations increasesA surprising result shows A surprising result shows that the synchronization that the synchronization regime becomes infinitely regime becomes infinitely wide, when the mass falls wide, when the mass falls below a special critical below a special critical value whose numerical value whose numerical value depends on the value depends on the shape of the vibrating shape of the vibrating bodybody

Critical Mass RatioCritical Mass Ratio

The synchronization regime of The synchronization regime of highhigh--amplitude vibration amplitude vibration ((shaded regimeshaded regime) extends to ) extends to infinite velocities as m infinite velocities as m approaches the value 0.54approaches the value 0.54

The lower plot, at infinite U*, The lower plot, at infinite U*, shows that there is a sudden shows that there is a sudden appearance of largeappearance of large--amplitude amplitude response when m* just falls response when m* just falls below 0.54below 0.54

Critical Mass RatiosCritical Mass Ratios

Tethered sphere system:Tethered sphere system:m* ~ 0.3m* ~ 0.3A pivoted cylinderA pivoted cylinderm* ~ 0.5m* ~ 0.5Elastic Mounted Cylinder in two degrees of Elastic Mounted Cylinder in two degrees of freedomfreedomm* ~ 0.52m* ~ 0.52

The Griffin PlotThe Griffin PlotSkopSkop--Griffin ParameterGriffin Parameter

S S –– StrouhalStrouhal NumberNumberm m –– massmassζζ -- damping factordamping factor

Griffin Plot is the plot of peakGriffin Plot is the plot of peak--amplitude parameter versus the amplitude parameter versus the SkopSkop--Griffin parameter as Griffin parameter as shown in the figureshown in the figureMy interpretation:My interpretation:Combining both the massCombining both the mass--damping and frequency into one damping and frequency into one plot, thereby studying their plot, thereby studying their combined effectcombined effect

Forced Vibration of a CylinderForced Vibration of a Cylinder

One approach to predicting VIV has been to One approach to predicting VIV has been to generate an experimental force data base by generate an experimental force data base by testing cylinders undergoing forced or controlled testing cylinders undergoing forced or controlled sinusoidal oscillations in a free streamsinusoidal oscillations in a free streamThe transverse force on bodies in harmonic, as The transverse force on bodies in harmonic, as well as multi frequency motion, is generally well as multi frequency motion, is generally decomposed into two components, one in phase decomposed into two components, one in phase with the velocity and one in phase with the with the velocity and one in phase with the accelerationacceleration

There are parametric regions where such There are parametric regions where such comparison is successful, and other comparison is successful, and other regions where the comparison is not closeregions where the comparison is not closeIn summary, despite the extensive work on In summary, despite the extensive work on controlled vibrations, it is still an open controlled vibrations, it is still an open question whether (strictly sinusoidal question whether (strictly sinusoidal motion) controlled experiments can be motion) controlled experiments can be used to predict free vibration accuratelyused to predict free vibration accurately

Effective Effective ElasticityElasticity

For systems with very less damping, the For systems with very less damping, the parameter k*, given by parameter k*, given by ––ωω**22m+k = k* is m+k = k* is called effective elasticitycalled effective elasticityThe inertial and the spring terms, which The inertial and the spring terms, which are precisely outare precisely out--ofof--phase are combined in phase are combined in thisthis

From these figures, if From these figures, if one chooses a one chooses a particular mass ratio particular mass ratio (m*), then one can (m*), then one can deduce the deduce the amplitude (A*) and amplitude (A*) and frequency (f*) as a frequency (f*) as a function of U* function of U* (velocity), and build (velocity), and build up a complete up a complete response plot for the response plot for the chosen m*chosen m*

XY Motion of XY Motion of BodiesBodies

A large number of papers are dedicated to the A large number of papers are dedicated to the problem of a cylinder vibrating transverse to a problem of a cylinder vibrating transverse to a fluid flow (Y motion)fluid flow (Y motion)Only a very few papers deal with bodies that Only a very few papers deal with bodies that are also allowed to vibrate inare also allowed to vibrate in--line with the flowline with the flowPrincipal question Principal question -- How does the freedom to How does the freedom to vibrate invibrate in--line with the flow influence the line with the flow influence the dynamics of the fluid and the structure?dynamics of the fluid and the structure?

Bodies in XY motion do not lead to surprising Bodies in XY motion do not lead to surprising changes in the expected maximum resonant changes in the expected maximum resonant amplitudes as compared to bodies in Y amplitudes as compared to bodies in Y motion.motion.One of the most interesting results from is that One of the most interesting results from is that even small amounts of stream wise motion even small amounts of stream wise motion ((AAxx/A/AYY = 20%) can inhibit the formation of the = 20%) can inhibit the formation of the 2P mode of vortex formation2P mode of vortex formationFullFull--scale piles in an ocean current, and scale piles in an ocean current, and similar cantilever models in the laboratory, similar cantilever models in the laboratory, vibrate invibrate in--line with the flow with peak line with the flow with peak amplitudes of the cantilever tipamplitudes of the cantilever tip

In most practical cases, In most practical cases, cylindrical structures (such as cylindrical structures (such as riser tubes or heat exchangers) riser tubes or heat exchangers) have the same mass ratio and have the same mass ratio and the same natural frequency in the same natural frequency in both the stream wise (X) and both the stream wise (X) and transverse (Y) directionstransverse (Y) directionsWhen m* < 6, there is new When m* < 6, there is new response branch with response branch with significant stream wise motion significant stream wise motion appears appears -- ““supersuper--upperupper””branchbranchThis yields massive amplitudes This yields massive amplitudes of three diameters peakof three diameters peak--toto--peak (A*peak (A*YY »» 1.5)1.5)This response corresponds This response corresponds with a new periodic vortex with a new periodic vortex wake mode, comprising a wake mode, comprising a triplet of vortices in each half triplet of vortices in each half cycle, defined as a cycle, defined as a ““2T2T””

Vortex Induced Vibration Vortex Induced Vibration Suppressors Suppressors

Various devices and Various devices and configurations are configurations are attached to pipe attached to pipe surfaces to break up, surfaces to break up, or suppress, the or suppress, the vortexvortex--causing causing currents currents

SuppressorsSuppressors

Suppressors can beSuppressors can beAlternating buoyant and bare joints of drilling riser in the higAlternating buoyant and bare joints of drilling riser in the high current h current area area Placing fairings or strakes on the outside of the drilling riserPlacing fairings or strakes on the outside of the drilling riser

•• STRAKESSTRAKES -- Three strakes, each made of the longThree strakes, each made of the long--lasting lasting SPLASHTRON, break up the current path, preventing eddies. The SPLASHTRON, break up the current path, preventing eddies. The strakes are bonded to the coating in a spiral fashion, with eachstrakes are bonded to the coating in a spiral fashion, with eachstrake making a full 360strake making a full 360°° turn around the pipe every 19 feet. The turn around the pipe every 19 feet. The strakes can be customstrakes can be custom--designed to fit particular riser requirements. designed to fit particular riser requirements.

SummarySummary

Fundamentals Fundamentals Different modes of VIV Different modes of VIV FullFull--scale data are inadequate for fluidscale data are inadequate for fluid--structure interactions in a variety of structure interactions in a variety of conditions, including sheared flows in the conditions, including sheared flows in the oceanoceanVIV SuppressorsVIV Suppressors

ReferencesReferences

Williamson. C. H. K and Williamson. C. H. K and GovardhanGovardhan. R, . R, ““Vortex Vortex Induced VibrationsInduced Vibrations””, Annual Review of Fluid , Annual Review of Fluid Mechanics 2004, 36:413Mechanics 2004, 36:413--5555Singh S. P and Singh S. P and MittalMittal. S, . S, ““VortexVortex--induced induced oscillations at low Reynoldsoscillations at low Reynolds’’ numbers: numbers: hysterisishysterisisand vortex shedding modesand vortex shedding modes””, Journal of Fluids , Journal of Fluids and Structures 20 (2005) 1085and Structures 20 (2005) 1085--11041104WikipediaWikipediawww.marktool.comwww.marktool.com

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