2-way fsi simulations on a shock absorber check valve839875/fulltext01.pdf · 2-way fsi simulations...

2-way FSI simulations on a shock absorber check valve

M.Sc. Degree Project

Tommy [email protected]

KTH Royal Institute of TechnologyDepartment of Aeronautical and Vehicle Engineering

Stockholm, Sweden

Supervisors:Matteo Pelosi

Ohlins Racing ABUpplands Vasby, Sweden

Georg Ahrberg

Ohlins Racing ABUpplands Vasby, Sweden

Examiner:Stefan Wallin

KTH Royal Institute of TechnologyDepartment of Mechanics

Stockholm, Sweden

January 7, 2015Stockholm, Sweden

Abstract

A component of a hydraulic shock absorber, a check valve, is analyzed usingboth numerical simulations as well as experimental testing. A fluid-structureinteraction (FSI) model is set up in ANSYS Workbench and is validated throughphysical experiments - both steady state and transient. The fluid field is solvedin ANSYS Fluent and the structural deformation is solved for in ANSYS Struc-tural. The coupling is made using ANSYS System Coupling.

The report covers the fundamentals of FSI analysis - methods of coupling fluidand structure solution fields and methods for adapting the fluid mesh to accountfor a changing geometry. A brief background on general moving/deformingmesh algorithms are presented but the emphasis lies on the methods availablein ANSYS Fluent and how to apply these on the case of a shock absorber checkvalve.

A moving/deforming mesh consisting of tetrahedral cells without inflation layerson wall boundaries proves the most robust dynamic mesh setup. The exclusionof inflation layers is shown to significantly affect the solution at low valve liftheight. At full lift the exclusion of inflation layers has no influence on the solu-tion. The check valve is 4-fold axisymmetric but is shown to exhibit asymmet-rical displacement. This is due to an asymmetrical fluid pressure distributionon the check valve.

Steady state FSI simulations show satisfactory correlation to flow bench exper-iments at low flow rates. The opening pressure differential of the check valve,determined by the spring preload, is accurately predicted by the FSI model. Atflow rates above 10 l/min the differential pressure is under predicted, due tosimplifications to the computational domain.

Transient simulations and experiments both show an oscillatory pressure differ-ential across the check valve as it opens, albeit with different frequencies.

Sammanfattning

En backventil tillhorande en hydraulisk stotdampare analyseras bade numerisktsaval som experimentellt. En FSI-modell (Fluid-Structure Interaction) stallsupp i ANSYS Workbench och valideras med fysiska experiment - bade statis-ka och transienta experiment. Stromningsfaltet beraknas i ANSYS Fluent ochstrukturdeformationer beraknas i ANSYS Structural. De bada losningar kopplassedan samman i ANSYS System Coupling.

Rapporten behandlar grundlaggande FSI - kopplingsmetoder och metoder forrorliga berakningsnat. Generella metoder beskrivs kortfattat men fokus liggerpa de metoder som finns att tillga i ANSYS Fluent, och hur dessa kan tillampasi analysen av backventilen.

Ett rorligt berakningsnat bestaende av tetraeder utan inflationlager pa vaggrandervisar sig vara det mest robusta dynamiska natet. Uteslutandet av inflationla-ger har dock betydande paverkan pa losningen vid laga ventillyfthojder. Vidfullt oppen backventil har avsaknaden av inflationslager forsumbar paverkanpa losningen. Backventilen ar 4-faldigt rotationssymmetrisk men oppnar andaasymmetriskt. Detta forklaras av en asymmetrisk tryckdistribution pa backven-tilen.

Statiska FSI-simuleringar visar pa tillfredstallande overenstammelse medflodesbanksmatningar vid laga volymsfloden. Backventilens oppningstryck pre-dikteras val av FSI-modellen. For volymsfloden over 10 l/min underbestamstryckfallet pa grund av forenklingar av CFD-domanen.

Bade transienta simuleringar och transienta experiment visar pa oscillationer itryckdifferentialen over backventilen, om an med olika frekvens.

Preface

This thesis lying in front of you is the final proof of competence for obtaining aM.Sc. degree in Aerospace Engineering from the Royal Institute of Technology(KTH), Stockholm. The project has been carried out at Ohlins Racing AB inUpplands Vasby, Sweden, under the supervision of Dr. Matteo Pelosi and GeorgAhrberg. Examiner is Dr. Stefan Wallin from the Department of Mechanics atKTH.

I want to thank my examiner Dr. Stefan Wallin for much appreciated feedbackand advice and I would like to thank my supervisors Dr. Matteo Pelosi andGeorg Ahrberg at Ohlins Racing for their support and guidance. I would alsolike to thank Tobias Berg, ANSYS Sweden, for valuable technical support andadvice throughout the project.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Ohlins TTX . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Check valve . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Project description . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Similar studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theory 11

2.1 CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 FSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Energy conservation . . . . . . . . . . . . . . . . . . . . . 14

2.4 Dynamic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Methods available in Fluent . . . . . . . . . . . . . . . . . 15

3 Method 17

3.1 Flow bench measurements . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Damper dynamometer tests . . . . . . . . . . . . . . . . . . . . . 20

3.3 Fluent setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.4 Turbulence model . . . . . . . . . . . . . . . . . . . . . . 29

3.3.5 Mesh study . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 38

3.3.7 Fluid properties . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Structural setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 System Coupling setup . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5.1 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5.2 Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Final transient model . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Results 45

4.1 Flow bench measurements . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Steady state FSI . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Dynamometer tests . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Transient FSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Conclusion and Discussion 53

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2.1 2-way FSI as a development tool . . . . . . . . . . . . . . 56

5.2.2 Suggestions for further studies . . . . . . . . . . . . . . . 56

References 59

Appendix A UDF’s and Profiles 63

Appendix B Model settings (360◦ case) 65

CHAPTER1Introduction

With computing power increasing exponentially with time according to Moore’slaw [1] the benefits of simulations as opposed to conducting physical experimentare increasing as well. However, an inaccurate model will not produce moreaccurate results by the use of more computing power. Here lies the scope of thisthesis - to set up and also validate a numerical simulation.

1.1 Background

Ohlins Racing has been developing high performance shock absorbers for thevehicle industry since 1976. The development teams rely both on simulationsand experimental testing in the development of new products. A hydraulicshock absorber works by the principle of controlling the oil flow rate through aset of flow restricting valves. In Ohlins TTX models the damping characteristicare determined by external valving and the flow path of the oil is controlledby check valves. These check valves have been shown to also influence thedamping characteristics. Figure 1.1 shows the damper force measured in adynamometer in an experiment done prior to the start of this degree project.The only difference is check valve setup, yet the damping characteristics aredifferent. This has inspired this FSI analysis of the check valve.

1.1.1 Ohlins TTX

A shock absorber of the conventional hydraulic type operates according to theprinciple of converting kinetic energy to heat through viscous dissipation ofenergy in a fluid. This is done by forcing a fluid through flow restricting valves,which results in shearing of the fluid. The internal work done by the fluidfield working against its own internal shear stresses produces heat which is thentransferred, via the cylinder and piston, to the surrounding air.

Typically a shock absorber consist of a piston on a piston rod running inside acylinder, sealed on the one end and sealed around the piston rod on the other

2 CHAPTER 1. INTRODUCTION

0 0.5 1 1.5 2

·10−2

−200

0

200

400

Time [s]

Forc

e[N

]

3 N/mm, preload 3.9 N


Figure 1.1: Comparison in measured damper force between twodampers with different check valve setups.

end. This is called a single tube damper. The cylinder is filled mostly withviscous damper fluid but also a portion of nitrogen gas in a chamber separatedfrom the oil by a floating piston. As the damper is compressed the piston rodwill displace more of the internal volume in the cylinder. With the damperbeing a closed hydraulic system, the internal volume of the system will thendecrease. Vice versa in rebound stroke, the internal volume of the damper isincreasing. This change in internal volume is taken up by the nitrogen gas, asit is of much higher compressibility than the damper fluid.

The fluid is driven through port holes in the piston whereupon stacks of elasticshims are placed (see Figure 1.2). These shims determine the opening pressureand the opening characteristics of the flow ports. Flow in both directions gothrough the piston. In compression stroke the rebound side shim stack is fullyclosed and in rebound stroke the compression side shim stack is closed. Thepressure differential across the piston is responsible for the damping force.

Refering to Figure 1.3, a TTX damper hydraulic scheme is shown. A twin-tube Ohlins TTX design presents many differences to a traditional single tubedamper. As mentioned earlier, on a conventional single tube damper the flowports and shim stacks are mounted on the piston. The pressure difference acrossthe piston drives the fluid through the valves in the piston. The twin tube TTXdamper works by to the same principle only the piston is solid and the valvesare external. Two sets of valves control compression and rebound dampingrespectively.

In compression stroke the fluid will flow through the compression side valveset and bypass the rebound valve set via a check valve. In rebound strokethe flow goes the other way i.e. through the rebound valve set and aroundthe compression valve set via another check valve. The advantage to this designwith the solid piston and external valves is that the gas reservoir is connected tothe low pressure side of the fluid at all times. Comparing this to a conventionaldamper where the gas reservoir is mounted on one side of the piston (typicallyon the compression side [2, p. 4] which is the side that has the higher pressurein compression stroke). This means that in compression stroke the flow throughthe piston is driven by the low pressure side, since on the compression side the

1.1. BACKGROUND 3

Cylinder

Pistonrod

External gasreservoir

Figure 1.2: A conventional single tube damper. Source: Ohlins Racing.

gas will compress and thus no there will be no significant pressure build-up inthe fluid.

The problem here is cavitation. A fluid that is subjected to large pressuregradients can, if the pressure is low enough, cavitate i.e. release gas bubbles ofair entrained in the fluid and further evaporate to gas state. When subjectedto higher pressure again, these gas bubbles implode generating shock waves,which can lead to damages on nearby surfaces. To reduce the risk of cavitationwhat is done is the gas pressure is set high enough to have the pressure on thedownstream side high enough to resist cavitation. On the TTX damper withits external valve the gas reservoir is connected to the downstream side both incompression and rebound stroke. The flow is then driven by the high pressureon the upstream side and the downstream pressure is always equal to the setgas pressure. With this design the gas pressure can be set lower than for theconventional single tube damper. The lower gas pressure also enables the useof looser fitting seals making for lower friction [2, p. 6].

1.1.2 Check valve

As stated above the damping in either direction is handled by two separate setsof valves in a TTX damper. One for compression and one for rebound. Thevalve set not in use is bypassed using a check valve. This is the hardware ofinterest in this M.Sc. degree project. Its function is to fully restrict flow in onedirection and open to let the fluid pass freely in the other direction. Of course,this component could also be designed to add additional damping, although its


Checkvalveopen

Checkvalveclosed

Checkvalveopen

Figure 1.3: Flow schematics of a TTX damper of through-rod type.Flow path in compression and rebound stroke. Source: Ohlins Racing.

main purpose is to control the flow path. The check valve is shown in Figure1.5.

(a) Seen from the upstreamside.

(b) Seen from the downstreamside.

Figure 1.4: Check valve.

The pressure difference across the check valve forces it to open and the fluidis allowed through. A flat shim, preloaded by a coil spring, rests on a seatsealing off four port holes. The seat is slotted to reduce contact area. In thecheck valve open state the fluid passes through the ports and through the gapformed between the shim and the seat in an outwards radial direction. Figure1.4 illustrates the design and function.

1.1. BACKGROUND 5

Flow path in mainflow direction

Shim

Coil springSpring collar

Lock ringFlow path in reverseflow direction

Figure 1.5: Check valve design and function (section view).

Throughout this report the fluid is referred to as upstream and downstreamof the check valve. Figure 1.6 shows the check valve mounted in a flow benchfixture with the upstream and downstream sides marked.

Inlet

Outlet

Upstream

Downstream

Figure 1.6: Check valve mounted in the flow bench fixture.

The check valve has a very low opening pressure compared to the dampingvalves. Its function is not to control flow rate and thus affect damping charac-teristics but solely to set the flow path of the fluid. However, as the check valveis not completely non-restricting to the flow some contribution to the damp-ing will come from this component. Transient experiments done in the pastat Ohlins Racing have shown oscillations in the pressure difference across thecheck valve. Figure 1.7 shows the oscillations in pressure across the check valveon the compression side i.e. the check valve that is open when the damper is inrebound stroke.

The shim is made out of the hardened and tempered carbon steel Sandvik 20C[3]. It is characterized by good flatness, surface finish and high fatigue strengthunder bending and impact stress. The same material is used for the shim stackswhere a lot of bending stress can occur. The shim in the check valve is unlikelyto see a lot of bending stress as the low preload of the coil spring allows fortranslation of the shim rather than bending.


0 0.5 1 1.5 2

·10−2

0.00

0.05

0.10

0.15

Compressionstroke

check valveclosed

Reboundstroke

Time [s]

∆p

[MP

a]

Oscillations inpressure differentialover check valve.

Figure 1.7: Dyno chart showing pressure drop over the check valvein the compression side cavity. Spring rate: 3 N/mm, preload 3.9 N,maximum lift height: 2 mm. Piston velocity is 50 Hz sinusoidal with amaximum amplitude of 0.1 m/s.

1.2 Fluid-structure interaction

The analysis of a system consisting of a fluid and a solid can often be simplifiedby making the assumption that the structural deformation is small and neg-ligible. In cases where this assumption is no longer valid, a coupled analysismight be imperative. Such is the case if the physical coupling between fluidand structure is strong e.g. in the analysis of a sail on a sailboat. Here thesail is the structure and the surrounding air is the fluid. The wind gives riseto aerodynamic forces on the sail and as it is not rigid, it deforms. As thesail is changing its shape the aerodynamic forces change as well, up until anequilibrium is reached (given that the wind is constant and steady). This is anexample where the physical coupling of fluid and solid is strong. A change influid flow gives rise to significant deformation of the structure and a change inthe geometry of the structure greatly affects the fluid flow.

In cases where deformations are small, the fluid-structure coupling is typicallyweak e.g. a rotating boat propeller immersed in water. The propeller will inducea jet of water behind it. The fluid is thus greatly affected by the structure. Notby its deformation but by its rigid body motion. The structure is affected bythe forces induced by the fluid. There is a deformation but it is small and theeffect on the flow field of this small deformation is negligible. This is an exampleof physically weak coupled interaction.

The interaction can be stable where the structural deformation will convergetowards a steady state deformation or it can be unstable. A good example of anunstable interaction is wing divergence. A poorly designed airplane wing willwhen deformed by the aerodynamic load increase its angle of attack and thusgive rise to increased load and in the worst case scenario break of completely.The World War I fighter aircraft Fokker-D8 suffered numerous accidents dueto this phenomenon [4, p. 2]. The interaction can also by dynamic in nature- dynamically stable or dynamically unstable. An interaction is dynamicallyunstable when the oscillations of the structure is increasing in amplitude. Think

1.3. PROJECT DESCRIPTION 7

of an airplane wing again. When standing still on the runway a disturbance tothe wing structure will cause a vibration that quickly settles due to the internaldamping of the structure and to some extent the damping coming from thesurrounding air. When at speed, an induced disturbance will still be damped bythe structure but combined with unsteady aerodynamics the aerodynamic forcefrom the surrounding air might now add to the amplitude of the disturbance.Even if the structure is statically stable it can still become dynamically unstableand the oscillations will grow uncontrollably [4, p. 4].

There are several ways of modeling fluid-structure interactions, depending onthe physical nature of the problem. In the case of a strong physical couplingthe numerical coupling must also be strong. If the physical coupling is weak itmight be sufficient to solve the problem numerically using a weak numericallycoupled approach which would be less computationally expensive.

The modeling of the fluid and the structure can be done in several ways, anddepends on the geometry of the structure and the complexity of the the fluidflow. Some cases can be successfully modelled using simplified linear models.The case of a long slender airplane wing is such an example, where the fluidforces can be modeled using potential flow theory and the wing structure can bemodelled using beam theory [4]. In the case of this M.Sc. degree project wherean opening and closing check valve is studied, the fluid flow is solved for usingfinite volume CFD and the structural deformation will be solved for using thefinite element method. The coupling of the two solution fields is done using apartitioned iterative 2-way method.

1.3 Project description

The goal of this M.Sc. degree project is to set up a robust and accurate 2-wayFSI model in ANSYS Workbench for simulating the transient behaviour of acheck valve. The solution is to be validated with flow bench measurements andshock absorber dynamometer tests.

A simplified domain is used for the majority of the work as this allows simula-tions to be run on a laptop computer. A final larger model, based on lessonslearned from the simplified model, is in the end stage of the project, run on acomputer cluster.

1.4 Similar studies

In 2003 Leon Leventhal published a study [5] on a ball check valve using bothexperimental methods and FSI. In the study the author is able to resolve thedynamic opening phase of the check valve as well as bouncing effects whenclosing. The FSI tool used is ADINA [6] which is a direct-coupled FSI code.The fluid and structural models are 2D axisymmetric. A small gap of 20 µmis left between ball and seat in its closed state to not pinch off the fluid mesh.In the structural model, the coil spring is modelled using a spring element.The paper does not include much detail about the CFD model or what method


is used to deform the mesh, and although experiment are conducted there isno direct comparison to the FSI model for validation. Figure 1.8 shows theaxisymmetric 2D fluid domain of the study.

Figure 1.8: CFD domain in Leventals study [5].

Parameter studies are done both experimentally and using the FSI model andboth studies lead to the same conclusion. Lower spring preload reduces theinitial pressure spike and oscillations in the opening phase but can cause ballbounce at closing. FSI studies showed that spring rate had minimal influencewhile experiments gave inconclusive results. The check valve in this study wasof ball-in-seat type. Reducing seat area showed lower pressure spikes and lessoscillations at check valve opening. This effect is explained as an effect of asmaller throttling area between ball and seat and thus less influence by Bernoullieffects. It was also shown that a ball with a diameter as small as the seat innerdiameter made for smaller pressure spikes.

Price et. al of Engineering Dynamics Inc, Texas [7] showed in a study of acheck valve used in a reciprocating pump the effects of modifying shim andseat geometry. The check valve is similar to the one studied in this M.Sc.degree project, a flat shim resting on a seat and preloaded by coil springs,albeit pressures and length scales are about one order of magnitude larger inthe study by Price et. al. Figure 1.9 shows the check valve of their study.They refer to the adhesion force between shim and seat in the opening phaseas stiction (short for static friction) and in their study they found the stictionforce to be a function of the width of the seating surface, the viscosity of thefluid and the initial film thickness which depends on the surface finish. Theymodel the stiction phenomenon by considering two flat plates immersed in afluid and as the two surfaces are separated fluid fills the void and in doing sogenerates a force on the surfaces in accordance with Bernoulli’s principle. Thestatic pressure in the high velocity fluid filling the void is lower than in thesurrounding fluid. In the case of their check valve this stiction phenomenon wasenough to cause cavitation leading to damage on the shim and seat surface. Toreduce seat area they machined grooves into the seat, which effectively reducedthe stiction force and also increased the flow rate of the pump the check valve

1.4. SIMILAR STUDIES 9

was used in. This was due to reduced ”valve lag” in the pump. It was alsoshown that a fluid with lower viscosity gave rise to smaller pressure spikes.

Figure 1.9: Check valve used in the study by Price et. al [7].

CHAPTER2Theory

The term FSI encompasses a number of different methods. The solution canbe steady-state or transient, the coupling can be weak or strong, the solid so-lution can be structural deformation, temperature or both. The fluid field canbe solved for using simple methods like potential flow, or computationally ex-pensive methods such as DNS. In this M.Sc. degree project the solid solutionis structural deformation solved for using the finite element method in ANSYSStructural - both static and transient. The fluid field is solved for using finitevolume CFD in ANSYS Fluent. The two solution fields are coupled using apartitioned iterative method in ANSYS System Coupling. Below follows a brieftheoretical background of the solution methods, and coupling method used.

2.1 CFD

Computational fluid dynamics (CFD) is the name for a number of methods whenit comes to solving a fluid flow field numerically. In common for most CFDmethod are the Navier-Stokes equations1. This set of equations can describeany fluid flow. There is yet no mathematical proof that a solution exists to theNavier-Stokes equations for arbitrary boundary conditions. This is part of theMillennium problem2. In practice however, with proof or not, the equationscan be solved. The equations govern the conservation of mass, momentum andenergy. In the case of finite volume CFD, the equations are expressed as volumeintegrals and solved on a finite volume mesh. The mesh resolution governsthe length scale of flow features that can be resolved. On turbulent flows,solving Navier-Stokes equations on a mesh fine enough to resolve the smallestlength scales, the chaotic fluctuations that is turbulence, is computationally veryexpensive and is yet only relevant for academic purposes. This would be, whatis known as DNS - Direct Numerical Simulation. Instead what is commonly

1There are also the Euler equations for inviscid flow and lattice Boltzmann equation usedin codes such as PowerFlow [8] and XFlow [9].

2One of the Millennium problems is to derive proof of existence and smoothness ofNavier–Stokes solutions for arbitrary boundary conditions. The prize for doing so is $1,000,000[10, p. 57].

12 CHAPTER 2. THEORY

done is to resolve mean velocities and model fluctuations, as done in RANS(Reynolds-Averaged Navier-Stokes) CFD.

In between DNS and RANS on a scale of computational cost lies LES (LargeEddy Simulation). In LES the smallest turbulent length scales (sub-grid scales)of the flow field are modelled while the larger length scales are resolved on thegrid. In RANS CFD, also the larger length scales are modelled.

The idea behind RANS is to separate the flow velocity into an average and afluctuating part. The fluctuating part, the Reynolds stress needs to be modelledto close the system of equations. The models for doing so are called turbulencemodels, and involve a statistical averaging procedure of the Reynolds stress [11].

The use of a turbulence model to model the smallest scales of the flow field andresolving the larger scales can be an effective way to solve a flow field. It ishowever important to use a turbulence model suited for the particular flow thatis studied, as no turbulence model is universal and suited for each and everyflow problem.

2.2 FEM

The finite element method (FEM) is a numerical method of approximating thesolution to a differential equation on a finite element grid. The method wasinitially developed for stress and displacement calculations for structural anal-ysis [12, p. 429] but is not limited to solid mechanics. The same discretizationmethod can also be used for solving thermal, fluid and electrostatic problems.In the scope of this thesis the use is limited to solid mechanics. The domainis divided into finite elements connected to each other in nodes. Each elementis represented by an algebraic equation relating stiffness and displacement toforce making for a system of equations to be solved for the whole structure.Depending on the type of element used different problems can be solved. Com-pare this to the finite volume discretization of finite volume CFD, where eachfinite volume acts as a control volume but does not include any details aboutthe physics of the fluid. In finite element analysis it is the choice of elementtype that govern the physics of the problem. For structural problems there areprimitive elements such as bar and beam elements applicable to simpler prob-lems as well as continuum elements such as plates and solids for discretizingmore complex problems.

2.3 FSI

As described earlier in the introduction, the physical coupling between solid andfluid can be weak or strong. So can the numerical coupling between the flowsolution and the structural solution. 1-way FSI is a weak numerical coupling.This is where data is transferred in only one direction. For example the flowsolution is calculated and sent to the structural solver which solves for a defor-mation, whereupon the flow solver calculates the solution for the next time stepand the process continues. This is applicable only to cases that are physically

2.3. FSI 13

weakly coupled, as in the example of boat propeller used earlier. Figure 2.1aillustrates the principle.

There are two types of strong coupling methods - monolithic and partitionediterative. Monolithic FSI, or direct-coupled FSI, means that both deformationand flow is solved directly using the same solver. Software such as ADINA[6] and COMSOL Multiphysics [13] have these capabilities. The monolithicapproach requires more storage but is better suited than the partitioned ap-proach to problems with large nonlinearities [14, p. 1442]. For large problems,however, the partitioned approach is often the preferred method [15]. Solv-ing complex structural problem using the monolithic approach is ”in generalcomputationally challenging, mathematically and economically suboptimal andsoftware-wise unmanageable” [15].

The partitioned iterative approach uses separate solvers - one for the flow fieldand one for the structural deformation. The solution is transferred in bothdirections. Figure 2.1b illustrates the principle. During each time step solutiondata is transferred back and forth between the solver in coupling iterations untilthe coupling step is converged, that is when the data transfer is low enoughto meet convergence criteria. This is 2-way FSI. The most popular solutionprocedure is referred to in [15] as the Conventional Serial Staggered procedureand goes as follows:

1. Transfer the motion of the FSI boundary in the structural domain to thefluid domain,

2. Update moving mesh in the fluid domain,

3. Advance in time and compute fluid solution,

4. Convert pressure and fluid stress into structural loads,

5. Advance in time and compute structural solution.

Fn

Sn

Fn+1

Sn+1

time step n time step n+ 1

(a) 1-way coupling.

Fn

Sn

Fn+1

Sn+1

time step n time step n+ 1

(b) 2-way coupling.

Figure 2.1: Coupling types. F is fluid solution and S is solid solution.

When making multiple coupling iterations per coupling step it is not necessaryto reach full convergence for the fluid solution in every coupling iteration, aslong as the final coupling iteration is converged. It can actually be a good ideato not converge the fluid solution fully in each coupling iteration as this willprovide some damping to the solution.


For weakly coupled systems one stagger loop (coupling iteration) per time stepmight be enough. This would be explicit iterative FSI, and would not be suitedfor cases of strong physical coupling, unless it is only the steady state solutionsthat is of interest. An explicit iterative coupling scheme does not achieve a fullfluid-structure coupling in each time step and is in general energy increasing andhence numerically unstable [16, p. 2-3]. Of the monolithic and the partitionedapproach, monolithic FSI is the strongest numerical coupling. Note that ifeach coupling step in the partitioned solution procedure is fully converged a fullcoupling between fluid and solid is achieved, and the solution should in theory(if the data transfer residuals are infinitesimal) be the same.

For this M.Sc. degree project the partitioned iterative approach is used, inthe form of ANSYS Fluent as the fluid solver and ANSYS Structural as thestructural solver. ANSYS System Coupling handles the data transfer betweenthe two solvers.

2.3.1 Energy conservation

An FSI system consist of two subsystems - a fluid system and a structural sys-tem. The boundary condition that couples fluid system to structural system isthe FSI interface. On this interface pressure loads from the fluid are transferredto the structural domain and causing a deformation. Vice versa, deformationsare passed back to the fluid system giving rise to forces on the fluid. As the twosubsystems together form a closed system, they share a global energy budget.As the FSI interface moves the solid performs work against the internal forcesof the fluid, producing a change in its energetic status [17, p. 342]. To studytransient effects in a fluid-structure interaction it is important that the couplingscheme is energy conservative. Especially when dealing with systems of unstablenature, such as in, for example, the study of wing flutter. When using FSI tofind limits of stability for such problems it is important to use a method thatis as globally conservative as possible to not influence the solution by adding ortaking away artificial numerical damping [15].

Michler et. al [16] compares, in a numerical experiment, solution accuracy andstability using conservative and non-conservative monolithic schemes. Theyshow how a scheme that conserves mass, momentum and energy at the fluid-solid interface results in a more accurate solution at the same computationalexpense, compared to a non-conservative scheme. They also show that violatingenergy conservation at the boundary can lead to numerical instabilities, evenfor a monolithic scheme.

In ANSYS System Coupling the force mapping between the two meshes on theFSI interface is locally (on element level) and globally conservative [18, lecture2, p. 13]. The two meshes, the fluid mesh and the solid mesh, does not need tomatch on the boundary. Typically the fluid mesh is finer than the solid mesh[19, p. 40], but it is recommended to have similar length scales on both meshesto maintain load resolution.

System Coupling does not make sure that the global energy of the fluid-structureis conserved - the scheme is non-conservative w.r.t. energy.

2.4. DYNAMIC MESH 15

2.4 Dynamic mesh

In FSI one important part of the problem setup is the dynamic mesh. As fora fixed geometry CFD-only calculation, the quality of the mesh is crucial inobtaining an accurate solution. In the case of FSI with transient boundaryconditions and moving boundaries, this becomes a sometimes difficult task. Inthis particular case of a check valve the geometry goes from a very narrow gaponly about 20 µm to a maximum gap of 1 mm in its open state. Meshing thesetwo geometries separately as fixed geometries is no difficulty. The difficultycomes in setting up a dynamic mesh that can adapt to the moving geometrywhile still maintaining good quality throughout the simulation.

There are two types of dynamic mesh methods. A moving mesh where the meshtracks and follow the moving boundary and deforms accordingly. The other typeis the non-moving (also known as overlapping or overset) mesh or interface-capturing method. With the interface-capturing method the moving boundaryis intersected with the overlapping mesh and the resolution at the boundary isdetermined by the overset mesh resolution and not the resolution of the actualmoving boundary. For all cases where a moving mesh is applicable the same isalso recommended rather than a non-moving mesh because of better accuracynear the FSI interfaces [20, p. 84]. Some cases are suited for a combined use ofboth methods. Such an example is water splashing where the bulk of the wateris tracked by a moving mesh while splashing droplets are captured using theinterface-capturing method [21].

2.4.1 Methods available in Fluent

In the case of the check valve, the geometry is simple enough to allow the use ofa moving mesh. Fluent has a few different ways of moving and deforming themesh. Available methods are Smoothing and Remeshing. Both methods belongto the moving mesh category described above. Smoothing essentially relocatesthe nodes to adapt to a moving boundary. The boundary nodes follow the mov-ing interface while the interior nodes are moved according to a chosen algorithm- Spring Smoothing or Diffusion Smoothing. Both Smoothing methods work ina similar fashion. Diffusion Smoothing is a bit more computationally expensivebut does tend to generate a better quality mesh and is recommended in caseswith large deformation and is therefore the method of choice in this case [22].There are benefits to using Spring Smoothing such as the possibility to have aprism layer mesh that moves rigidly with the moving boundary. More on thisin following chapter.

Remeshing works, in Fluent, by a set of size and skewness constraints, whensmoothing the mesh stretches or compresses cells too much or cells reach askewness limits, local remeshing is activated. Here nodes are reconnected. Cellsare either split or merged. When adapting the existing mesh using Smoothingand Remeshing is no longer possible while maintaining size and skewness con-straints a total remeshing of the moving mesh is triggered. This is in Fluentcalled Cell Zone Remeshing and creates a new mesh from scratch using the cellsize of the existing mesh. Usually when meshing a geometry such as this, local


refinement are made where velocity gradients are high and a coarser resolutionis used in non-critical areas. When using Remeshing, both local and the globalCell Zone method, the meshing algorithms work best on a mesh of uniform el-ement size. Local Remeshing works by set size limits and Cell Zone Remeshingcalculates the element size for the new mesh from the existing mesh. Thus hav-ing local refinements will affect the average element size, and will thus make foran increased number of elements after every Cell Zone Remeshing.

With Smoothing all types of elements can be used, while Remeshing worksonly with tetrahedral elements. The global Remeshing algorithm Cell ZoneRemeshing has the capability of remeshing both tetrahedrons and prisms. Thisis however a computationally expensive method compared to local Remeshing,and should thus only be used when local Remeshing fails to maintain size andskewness criteria.

There is also a third type of dynamic mesh method - Layering. This method iscommonly used when modelling in-cylinder flows in combustion engines. Thedownside is that it can cause instabilities when used on strongly coupled FSIcases such as this check valve case. When the FSI interface moves mesh layerswill be added or subtracted at the interface boundary, and this might cause errorto amplify and cause instabilities within the coupling step. For the purpose ofthe check valve setup modelled in this thesis the Layering method is deemedunfit because of this possible instability issue.

CHAPTER3Method

The end goal is to have a working FSI model for simulating transient check valvebehaviour. This model is set up and validated in steps. First a steady statemodel is compared to quasi-steady state flow bench measurements and latera transient model is set up and compared to transient damper dynamometertests. This chapter documents the setup and and execution of experiments andmodel.

3.1 Flow bench measurements

The flow bench consist of a pump driving damper fluid through a set of pipesleading up to a chamber where the specimen to be measured is placed. Figure3.1 shows the flow bench and Figure 3.2 the check valve fixture. The check valveis mounted in a fixture with the inlet being on the side and the outlet on thebottom. Pressure sensors are mounted upstream and downstream of the fixtureas shown in Figure 3.1.

Pressure sensor,downstream

Pressure sensor,upstream

Figure 3.1: Picture of the flow bench used for steady state experiments.

The flow bench can be configured to run in both directions. In this configuration

18 CHAPTER 3. METHOD

Figure 3.2: Fixture for mounting the check valve in the flow bench.Inlet is on the side and outlet at the bottom. The top is sealed after thecheck valve has been positioned.

the upstream pressure is measured in the large chamber holding the fixture. InFigure 3.3, a schematic overview of the flow bench is shown. Because of the largesize of the measurement chamber, the flow close to the upstream pressure sensoris assumed stagnant and the measured pressure is seen as the total pressureupstream of the check valve. The pressure sensor downstream of the checkvalve is connected to the outlet piping downstream of the fixture outlet. Thepressure here can not be assumed as the total pressure.

Pump

Measurementchamber

Check valvepup

pdown

Figure 3.3: Flow bench schematic.

According to Bernoulli’s principle total pressure is

p0︸︷︷︸total

pressure

= p︸︷︷︸static

pressure

+1

2ρU2︸︷︷︸

dynamicpressure

+ ρgz︸︷︷︸pressurelevel

(3.1)

where p is static pressure, ρ is fluid density, U is velocity, g is acceleration due togravity and z is elevation. The last term can be omitted in this case. To relateit in numbers, the maximum velocity (cross-sectional average) at the fixtureoutlet is

Vmax =Qmax

Aout= 4.9 m/s (3.2)

where Qmax = 7.5 · 10−4 m3/s (45 l/min) is the maximum flow rate used inmodel validations later, Aout = 1.5 · 10−4 m2 is the area of the fixture outletmaking for a dynamic pressure of

pdyn =1

2ρV 2

max = 0.010 MPa (3.3)

3.1. FLOW BENCH MEASUREMENTS 19

where ρ = 843 kg/m3 is fluid density at 25◦C. The dynamic pressure is notnegligible considering the order of magnitude of the measured pressure drop atthe same flow rate is O(0.1 MPa).

The measured pressure drop is the difference in total pressure upstream andstatic pressure downstream,

∆p = pin − pout + ∆pcalib.error = pin0 − pout (3.4)

since the flow upstream is assumed stagnant.

The pressure drop across the check valve is measured at different flow ratesto produce what is know as a p − Q diagram - steady state pressure dropagainst volumetric flow rate. Flow rate is increased linearly with time while thepressure upstream and downstream of the check valve is measured. The flowrate is increased at a rate slow enough to omit transient effects. By measuringpressure drop both on flow rate ramp-up and ramp-down and comparing thedata quasi-steady state behaviour can be confirmed.

At zero flow rate there is a measured pressure drop, probably due to a calibrationerror (∆p 6= 0 at Q = 0). The following measurements are therefore correctedwith this calibration offset shown in Figure 3.4.

0 2 4 6 8 10 12

−0.5

0.0

0.5

1.0·10−2

Calibrationerror

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a]

Figure 3.4: pref vs. volumetric flow rate for one of the tested checkvalve setups.

For the purpose of validating the CFD model alone a series of measurement aremade without the coil spring present, but with different maximum lift heights.Without the coil spring the shim lift will be equal to its maximum permittedlift for the full duration of the flow rate ramp. With the lift height knownfixed geometry CFD cases can be set up to validate the CFD model alone.Two spring collars of different lift height, 1 mm and 2 mm, combined with thinspacers allows the lift height to be changed in small increments. The volumetricflow rate is ramped from zero up to 45 l/min over a period of 30 seconds. Theflow rate is then decreased at the same rate down to zero. With this rate ofchange steady state behaviour is confirmed as the the p−Q relation is the samein ramp-up as ramp-down as shown for one of the check valve setups in Figure3.5.

To validate the FSI model measurements are made with the coil spring installed.


0 10 20 30 400.00

0.05

0.10

0.15

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a] Ramp-up

Ramp-down

0 10 20 30 400.00

0.05

0.10

0.15

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a] Ramp-upRamp-down

0 10 20 30 400.00

0.05

0.10

0.15

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a] Ramp-upRamp-down

Figure 3.5: ∆p vs. volumetric flow rate in ramp-up and ramp-down forthe case of a 3 N/mm spring and 3.9 N of preload.

Two different spring rates are tested in combination with the two spring collarsgiving a total of four setups (Table 3.1). There is also the possibility of changingspring preload. This option is left out for now.

Table 3.1: Flow bench test setups.

Spring rate [N/m] Preload [N] Max. lift [mm]Flow bench setup 1: 3 3.9 1Flow bench setup 2: 8 10.4 1Flow bench setup 3: 3 3.9 2Flow bench setup 4: 8 10.4 2

3.2 Damper dynamometer tests

In transient damper dynamometer tests on a motorcycle front end, carried outbefore the start of this M.Sc. degree project, oscillations in the pressure dif-ference across the check valve could be seen (Figure 1.7). If these pressureoscillations are effects of the check valve or effects of other flow restrictions inseries is hard to say. In an attempt to isolate the effects of the check valve another test is conducted. This time with an empty check valve and no shim stackin the rebound cavity, and pressure drop measured over the compression sidecheck valve.

For this test a damper dedicated for testing, with outlets for pressure sensors,is used (Figure 3.6). Driving the fluid is the piston and cylinder of a TTX36shock absorber. Specifications for the experiment damper are listed in Table3.2.

In rebound stroke the fluid passes through the empty valves on the reboundside, over to the compression side cavity and through the check valve. Pressureis measured upstream and downstream of the compression side check valve. Thesensors are mounted flush with the flow boundaries and the cross-sectional areais the same at the three sensor locations. Figure 3.7 shows the location of the

3.2. DAMPER DYNAMOMETER TESTS 21

Figure 3.6: Dyno damper.

Table 3.2: Dyno damper dimensions.

Piston diameter [mm]: 36Piston rod diameter [mm]: 12Valve cavity inlet/outlet diameter [mm]: 14

pressure sensors. Only two of the pressures are of interest here - upstream anddownstream of the check valve when the shock absorber is in rebound stroke.As both inlet and outlet to the check valve cavity are of the same cross-sectionalarea there will likely not be any difference in dynamic pressure due to differentflow velocities. The measured difference in static pressure is therefore assumedequal to the total pressure drop,

pindyn = poutdyn ⇒ ∆p = pin − pout = pin0 − pout0 (3.5)

Damping valve

Pressure sensorupstream

Check valve

Pressure sensordownstream

Empty dampingvalve

Gas reservoir

Empty checkvalve

Flow direction inrebound stroke

Figure 3.7: Dyno damper end piece (section view).

Three check valve setups are tested, listed in Table 3.3. Preload is changed byplacing shims between the lock ring and spring collar. In Setup 2 the spring ispreloaded with two of these shims (0.3 mm thick each). The maximum lift is setby the spring collar. To not make the check valve bottom out in Setup 2 with0.6 mm of added shims the lower spring collar (2 mm maximum lift) is usedfor this setup making the maximum lift 1.4 mm. The difference in maximumlift between Setup 1 and Setup 2 should have no influence as the check valve isunlikely to open fully. Steady state flow bench tests indicate that the check valve


in its most compliant setup (3 N/mm spring rate and 3.9 N of preload) reaches1 mm lift at about 30 l/min and the maximum flow rate in the dynamometertests that will be looked at is about 15 l/min. The check valve lift might exhibitsome overshoot in the transient tests and it is not impossible that the checkvalve reaches maximum lift, but it is unlikely. If so, it should be visible in themeasured pressure drop vs. time diagram.

Table 3.3: Dynamometer test setups.

Spring rate [N/mm] Preload [N] Max. lift [mm]Dyno setup 1: 3 3.9 1.0Dyno setup 2: 3 5.9 1.4Dyno setup 3: 8 10.4 1.0

The test case used for FSI model validation is a sinusoidal piston velocity withfrequency 50 Hz and maximum piston velocity 0.25 m/s. In earlier tests this iswhere oscillations in pressure were most apparent. The dynamometer has sen-sors for logging position. Figure 3.8 show the position and velocity (calculatedfrom position) for the test program used for FSI validation. Velocity is definedpositive in compression stroke and negative in rebound.

0 0.5 1 1.5 2

·10−2

−1.35

−1.30

−1.25

−1.20

−1.15·10−2

Com

p.

Reb.

Time [s]

Pos

itio

n[m

]

(a) Dyno position.

0 0.5 1 1.5 2

·10−2

−0.250

−0.125

0.000

0.125

0.250

Time [s]

Vel

oci

ty[m

/s]

(b) Dyno velocity.

Figure 3.8: Dynamometer states. Dyno position is measured whilevelocity is derived from position and time.

3.3 Fluent setup

The end goal is to set up a robust and accurate transient FSI model. As FSI is ingeneral computationally expensive, compared to solving a standalone fluid caseor a standalone structural case, it is worth-wile to simplify the case if possible.At least so during the setup and debug phase. Here a smaller simplified domainis used for the larger part of the work when investigating mesh size, dynamicmesh settings, turbulence modelling and time step size. In the final stage of theproject the size of the domain is increased to better represent the real geometryused in experiments.

3.3. FLUENT SETUP 23

3.3.1 Geometry

For studying the impact of mesh size, turbulence model, time step etc. the do-main is simplified to allow the case to be run on a laptop computer with reason-able computation times (overnight). The geometry upstream and downstreamof the check valve is that of the fixture used for the flow bench measurements,with the difference that the inlet is now on the same axis as the outlet makingthe geometry 4-fold axisymmetric. The outlet is extended to avoid reversedflow at the boundary, which can cause convergence issues. Figure 3.9 showsthe actual geometry of the flow bench fixture and the simplified geometry. Toreduce the size of the domain further only a 45◦ slice of the geometry is used ascomputational domain.

(a) Flow benchfixture geometry(section view).

(b) Simplifiedgeometry withextended outletand axisymmetricinlet (sectionview).

Inlet

Outlet

(c) 45◦ slice.

Figure 3.9: CFD geometry.

The CFD domain for the FSI case does not include the entire length of pipingin the flow bench leading up to the downstream sensor. The CFD domain isalso simplified upstream and does not include the large cavity where the up-stream pressure sensor is mounted, but any pressure drop between the upstreampressure sensor and the fixture inlet is assumed small and negligible.

The same geometry is used for the steady state simulations as for the firsttransient simulations.

3.3.2 Mesh

The fluid domain is separated into smaller subdomains (Figure 3.10) to createsweepable bodies where applicable, as the number of nodes and cells for a sweptbody is typically much smaller than ones meshed with the default mesher [23, p.231]. The regions connecting the swept regions are meshed with unstructuredtetrahedrons as a sort of buffer zones, connecting the swept mesh zones. Theregion surrounding the moving shim is separated from the rest of the mesh.


This is to make the size of the dynamic mesh domain as small as possible forcomputational reasons. The dynamic mesh operations takes place on a singlecore and can be a bottleneck in the solve process when the case is scaled upon multiple cores. Care is therefore taken to make the dynamic mesh region assmall as possible.

Inlet

Outlet

(a) Simplified domain.

Swept hex.mesh

Deformingmesh zone

Unstructuredtet. mesh

(b) Mesh zones.

Figure 3.10: Simplified geometry and mesh.

Dynamic mesh

Maybe the most time consuming and difficult part of setting up this FSI modelis the dynamic mesh. First of all, the fluid mesh needs to be continuous thusa small gap that can fit at least one cell needs to be left between the shim andthe seat. A 20 µm gap is left. Similar work done on a ball check valve showedthat this was small enough to capture Bernoulli effects1 [5].

Ideally the mesh would include inflation layer along all wall boundaries to betterresolve the boundary layer. In the initial narrow gap these prism cells wouldbe very thin. As the check valve opens the prism cells would ideally stretch, oradditional layers of prism cells would be added. When it comes to cell size, themesh would ideally be finer in regions with large gradients in pressure and/orvelocity and around boundaries with small radius of curvature and could bemade coarser in less active flow regions. In the case of fixed geometry CFD thisis usually not a problem, but when dealing with a transient FSI case with largedeformations compromises in mesh quality might have to be made.

As mentioned in the Theory section, there are two methods available in Fluent,applicable to this case, for adapting the mesh - Smoothing and Remeshing. Thegoal is to have a dynamic mesh that is robust and can handle going from a verynarrow gap to fully opened.

When investigating different types of dynamic mesh setups a prescribed motionis assigned to the shim using a Fluent Profile (Appendix A). The mesh deforma-tion can then be previewed without solving for the flow field. This proved to bea very useful feature during the setup. Both axial as well as some radial motion

1At check valve opening the fast flowing fluid in the gap gives rise to a low static pressureon the shim which delays check valve opening.


of the shim was tested. As a simplified problem the shim can be constrainedto axial motion only, but the ambition is to eventually replace the ideal springelement in the structural model with a model of the actual coil spring whichwill most likely lead to some radial motion as the reaction forces in a coil springare usually not perfectly axial.

Initially simple 2.5D2 cases where studied to get an idea of the influence of thedifferent Smoothing and Remeshing parameters, but this approach was quicklyabandoned for 3D cases since the dynamic meshing works differently on facesand interior volumes. Many different approaches were tried and a brief summaryof each approach follows below.

Dynamic mesh approach #1 - Unstructured tetrahedral mesh withinflation layers in the same domain:

This first mesh attempt consist of inflation layers along all boundaries and atetrahedral mesh in the rest of the deforming zone. The inflation layers weremade very thin in the gap. The problem is that the prism cells, which are alot smaller in size than the tetrahedrons is included in the cell size calculationdone by the Cell Zone Remeshing algorithm and thus for every time Cell ZoneRemeshing is triggered the cells will decrease in size and in the end the mesh willbe very dense. Figure 3.11 show the mesh after Cell Zone Remeshing has beentriggered. Initially the interior cells had the same dimensions as the face mesh(in blue). Disabling Cell Zone Remeshing and relying only on local Remeshingwas tried but caused problems of negative cells.

Small cells

Figure 3.11: Mesh after Cell Zone Remeshing. Inflation cells and tet.cells in the same cell zone make for an increase in number of cells everytime Cell Zone Remeshing is triggered.

An other issue is the splitting of the cells in the shim-to-seat gap. If the checkvalve opens too fast i.e. the time step is too large, the cells in the gap tendto stretch and become larger than the size constraints set. Figure 3.12 showsstretched cells in the gap. The mesh shown in the figure is in 2.5D i.e. onlyone element deep. On a 3D mesh the effect of stretched cells is not nearly asprominent. This goes to show how the dynamic mesh works differently on facecells and interior cells.

2A 2D mesh extruded one cell deep is usually referred to as 2.5D.


Stretched cells

Figure 3.12: Stretched cells in gap. Here on a 2.5D mesh.

In the early stages of exploring dynamic mesh settings, the stretched cells ap-peared to be a problem also on interior cells of a 3D mesh. By manually trig-gering Cell Zone Remeshing at predefined times as the check valve opens thestretched cells were be forced to split. However, after running the FSI case andmonitoring mesh deformation this method of manually forcing the cells in thegap to split proved unnecessary. The short time step used for the transientsimulation eliminated the problem.

Dynamic mesh approach #2 - Structured mapped mesh:

A mesh of hexagonal cells is mapped as shown in Figure 3.13a. Using a mappedhexagonal mesh precludes the use of Remeshing. Instead the mesh adaptationmust rely solely on Smoothing. Good mesh quality is kept in the gap, butthe problem is skew cells around the shim edges. After trying a multitudeof Smoothing settings, Spring based and Diffusion based, without success thisapproach was ruled out. With both Smoothing methods negative cells wereencountered at the shim edges as shown in Figure 3.13b.

(a) Undeformed mesh. (b) Negative cells.

Figure 3.13: Dynamic mesh approach #2. Mapped hexagonal mesh.

Dynamic mesh approach #3 - Mapped mesh in gap and unstructuredtetrahedrons in the rest of the domain:

To have good mesh resolution in the gap while still having a coarser and alsomore robust dynamic mesh in the rest of the domain, the domain is separated


into four subdomains. The region between shim and seat is mapped with hexag-onal cells while the rest of the domain is meshed with unstructured tetrahedrons.Smoothing is used in the hexagon regions. Since no cells will be added as thegap increases the number of hexagonal cells must be enough to provide satis-factory resolution also in the fully open state. Smoothing in combination withRemeshing is used in the tetrahedron regions. The nodes are unmatched onthe interface, as matching the cells would cause very small tetrahedral cells(which works poorly with Remeshing). The deformation of the tetrahedral cellsis satisfactory. The uniform cell size of the tetrahedral mesh makes both localRemeshing and Cell Zone Remeshing work to satisfaction. The reason for aban-doning this approach is the restriction in degrees of freedom of the shim. Thedynamic mesh work well for axial-only translation, while radial translation pro-duces skew hex cells at the mesh zone interface. Figure 3.14 shows the deformedmesh.

(a) Deformed mesh.(b) Closeup of skew cells atinterface.

Figure 3.14: Dynamic mesh approach #3. Hexagonal mesh in the gapand unstructured tet. cells in the rest of the dynamic mesh zone.

Dynamic mesh approach #4 - Unstructured tetrahedral mesh withinflation layers in separate domains:

This mesh is similar to approach #1 but now with the inflation layers in theirown separate cell zones (Figure 3.15). This makes the Remeshing of the tetra-hedrons work better. The problem with Cell Zone Remeshing encountered inapproach #1 is now gone. That is the problem with an underestimated cell sizefor the tetrahedral cells by the Cell Zone Remeshing algorithm caused by havingthe the prism cells in the same zone. The uniform tetrahedral mesh adapt toshim displacement in a robust manner (no negative cells). The prism cell zoneon the seat is seen by Fluent as a stationary zone. The prism layers on theshim is set as rigid body and assigned a ”dummy” motion. This is a sort ofworkaround in Fluent to allow the inflation layers to follow the shim boundaryand deform with it while not being part of the deforming zone. The inflationlayer motion is defined using a UDF (Appendix A) which in this case is empty.That is no motion is assigned, yet Fluent sees the inflation layers as a movingrigid body. This enables the feature ”Deform Adjacent Boundary Layer withZone” (works only with the Spring Smoothing method).

The problem with this mesh lies in the fact that the case is started from a verynarrow gap. The inflation layers are very thin and as the gap is increased the


(a) Note the large difference insize between the utmostinflation layer and the firsttetrahedrons.

(b) Stationary inflation layeron seat and moving inflationlayer on shim.

Figure 3.15: Dynamic mesh approach #4. Tet. cells and prism cells inseparate cell zones.

prism cells remain very thin compared to the rest of the mesh. The very thininflation layers in the gap cause a large size difference in the transition to thetetrahedral cells which led to convergence issues. This approach could be usedwith success if starting from larger gap, allowing for larger prism cells. Theambition of this thesis is to set up an FSI case capable of simulating an entireopening and closing cycle in the check valve, starting from a closed check valve.Therefore this approach is rendered unsuited for the task.

Dynamic mesh approach #5 - Unstructured tetrahedral mesh of uni-form cell size:

An unstructured tetrahedral mesh (Figure 3.16) of uniform cell size withoutinflation on the boundaries was found to be the most robust alternative. Itallows the simulation to be started from an almost entirely closed gap and itmakes both local Remeshing and Cell Zone Remeshing work well.

Figure 3.16: Unstructured tetrahedral mesh without inflation layer.Here at 0.1 mm lift.


3.3.3 Discretization

Only the swept meshes on inlet and outlet are aligned with the flow. Theunstructured tetrahedral mesh zones are not (a tetrahedral mesh is never alignedwith the flow) and in the deforming zone no boundaries have inflation layersfor good boundary layer resolution. For meshes of poor quality such as thismesh, especially in the deforming zone, the Coupled pressure-velocity couplingis recommended [24, p. 1464] and used for this case. Spacial discretization issecond order for all quantities to keep discretization errors to a minimum.

When remeshing and interpolating the solution onto a new mesh the temporaldiscretization is first order accurate. Using only Smoothing for the dynamicmesh would allow for a second order transient formulation. As the dynamicmesh uses both Smoothing and Remeshing in every time step the solution willbe first order accurate in time regardless of the transient formulation setting setin the Fluent solver input.

3.3.4 Turbulence model

The choice of turbulence model is dependent on the type of flow field to solve foras no turbulence model is universal. In industrial CFD, two-equation turbulencemodels are the most widely used [25, p. 698] and within this class of modelsk − ε [26] and k − ω [27] are the most common.

Both models exist in a multitude of variants. Realizable k − ε [28] have shownimprovements over the standard k − ε model where the flow exhibits strongstreamline curvature, vortices and rotation [24, p. 52] and is, in Fluent, therecommended model from the k − ε family. RNG k − ε [29] has also shown thesame improvements over the standard model. From the k− ω family of modelsthe SST k − ω [30] is recommended in Fluent [25, p. 699].

To gauge the influence the choice of turbulence model has on the solution, acomparative case is set up. The model is later to be compared to flow benchmeasurement with flow rates up to 45 l/min. The most turbulence is expected atthe highest flow rates and this is the case used for comparison. In this operatingpoint the check valve will be fully open (as is seen in flow bench results presentedlater). To determine whether the solution is turbulent or not levels of turbulentkinetic energy and turbulence intensity are looked at.

The comparison is made on a mesh with inflation layers on all wall boundaries(Figure 3.17). Although the dynamic mesh for the transient simulations willnot include inflation layers, the mesh for this comparison have been equippedwith such. The goal here is to study the difference between a turbulent vs. alaminar solution and not the impact of grid resolution. Therefore, to make theturbulence models perform as well as possible, the mesh is including inflationlayers. Note that with these inflation layers using the dynamic mesh approach#4 described earlier the shim would only be able to close to a minimum gap ofabout 0.4 mm. The three turbulence models are used together with EnhancedWall Treatment (EWT) which makes the turbulence models less sensitive tonear-wall mesh resolution. EWT uses enhanced wall functions3 in coarse near-

3Enhanced wall functions in Fluent blends the linear wall law for the laminar viscous


wall mesh regions and in regions where the mesh is fine enough to resolve theflow all the way down to the viscous sublayer (y+ ≈ 1) a two-layer model isused [24, p. 122].

Figure 3.17: Mesh used for turbulence model comparison.

Figure 3.18 show y+ values for one of the turbulent solution. The values are inthe same range for all of the three tested turbulence models. y+ is a dimension-less wall distance defined as [31, p. 71]

y+ ≡ ρu∗y

µ(3.6)

where ρ is density, y is height of first cell, µ is dynamic viscosity and u∗ isfriction velocity defined as,

u∗ ≡√τwρ

(3.7)

where τw is wall shear stress,

τw = µ

(∂u

∂y

)y=0

(3.8)

Standard log-law wall functions typically work very poorly for y+ < 15. Forthis mesh where y+ < 3 the use of EWT is a necessity.

Figure 3.20 shows turbulent kinetic energy k plotted on a plane going throughthe center of the geometry (Figure 3.19). Turbulent kinetic energy is the kineticenergy in the modelled velocity fluctuation defined as [12, p. 563],

k =1

2u′iu′i =

1

2

(u′2x + u′2y + u′2z

)(3.9)

where q is the average of some quantity q. The highest levels of turbulent kineticenergy is found in the shear layers around the jet downstream of the check valve.Both k− ε models produce similar results while the k−ω model show only halfthe magnitudes of the k − ε models.

Velocity magnitude contours for the three turbulent solutions and the laminarsolution are shown in Figures 3.21. The flow field looks similar for all four cases

sublayer and the logarithmic wall law for the turbulent outer region using a y+ dependentblending function [24, p. 124].


Figure 3.18: Contours of y+ for Realizable k − ε with Enhanced WallTreatment.

Figure 3.19: Plane for plotting solution data.

with the difference that the jet extends further downstream for the laminarsolution than for the turbulent solutions. For the three turbulent solution thehigh velocity in the jet dissipates more quickly than for the laminar solution.

Reynolds number

When estimating whether a flow is laminar, turbulent or in transition, a usefultool is the Reynolds number (Re) - a non-dimensional number relating inertialforces to viscous forces in a fluid.

Re =ρUl

µ(3.10)

where ρ is fluid density, µ is dynamic viscosity, U is characteristic velocity andl is characteristic length [12, p. 146]. The choice of characteristic length andvelocity is dependent on the type of flow analysed.

High Re typically indicates turbulent flow while a low Re is an indication oflaminar flow. For flows inside or around more common geometries such as flowin pipes, wide ducts or flow around airfoils, the flow can be characterized bycomparing Re to empirically found transition Re. The flow in the check valvecan not as easily be characterized. However, the flow can be split up in smallerregions which can then be looked at individually. The flow at inlet and outlet is


(a) k − ω - SST. (b) k − ε - RNG.

(c) k − ε - Realizable.

Figure 3.20: Turbulent kinetic energy plotted on a plane through thedomain center.

thought of as pipe flow, and characterized using pipe flow transition numbers.For this Reynolds number study, the geometry of the dynamometer damperis used (as shown in Figure 3.7) where the inlet and outlet of the check valvecavity have the same cross section (14 mm in diameter). Two cases are used forcomparison - 0.1 mm lift and 1.0 mm lift with low and high flow rate.

Characteristic length for the inlet is the hydraulic diameter,

DH ≡ 4 · Cross-sectional area

Wetted perimeter(3.11)

which in the case of the inlet is,

DH,inlet = 4 · πD2/4

πD= D (3.12)

where D is inlet diameter.

Characteristic velocity is the mean velocity,

U =Q

A(3.13)

where Q is volumetric flow rate and A is cross-sectional area.


(a) k − ω - SST. (b) k − ε - RNG.

(c) k − ε - Realizable.(d) Without turbulencemodelling.

Figure 3.21: Contours of velocity magnitude ([m/s]) on center plane.

The flow in the gap is also characterized using the hydraulic diameter and meanflow velocity. The hydraulic diameter of the circumferential cross section is

DH,gap = 4 · hπDseat

2πDseat= 2h (3.14)

where h is lift height of the shim and Dseat is the diameter of the utmost seatingsurface.

Table 3.4 lists the Reynolds number in two operating points - low lift and lowflow rate and maximum lift and high flow rate.

For flow in straight pipes, most experiments show that transition to turbulencetakes place at a Reynolds number of about Re = 3000 [12, p. 517]. If comparingthe flow at the check valve inlet and the flow in the shim-to-seat gap to straightpipe flow, the check valve flow would appear laminar even at a high flow rate.

However, Reynolds numbers for transition to turbulent flow in straight pipes arenot directly comparable the check valve as the geometry of the check valve ismore complex. Also, most of the turbulence is likely to be produced in the shearlayers around the jet exiting the gap. Something that is not considered whencalculating the Reynolds number as done above. These difference aside, theReynolds numbers calculated here indicate low turbulence levels. Aside fromthe turbulence produced in the jet that is.


Table 3.4: Reynolds numbers.

• Operating point 1: 0.1 mm lift, ∆p = 0.05, Q = 4.0 l/min,

• Operating point 2: 1.0 mm lift, ∆p = 0.15, Q = 41.1 l/min.

Char. length DH [m] Char. vel. U [m/s] ReOperating point 1:Inlet: 1.67·10−2 0.43 104Gap: 2.00·10−4 10.94 108

Operating point 2:Inlet: 1.67·10−2 4.45 1073Gap: 2.00·10−3 11.24 1115

Turbulence intensity

A way to measure the level of turbulence is to relate the average fluctuatingvelocity to a reference mean velocity [31, p. 64]. Turbulence intensity is definedas

T.I. ≡ u′

Uref(3.15)

where u′ is the root mean square of the modelled turbulent velocity fluctuations,

u′ =

√1

3

(u′2x + u′2y + u′2z

)=

√2

3k (3.16)

Uref is chosen as the velocity at the center line of the jet (Uref = 18 m/s)exiting the shim-seat gap as done in [32] and [33]. Figure 3.22 shows turbulenceintensity for the three turbulent solutions.

Referring to Figure 3.22, the maximum turbulence intensity is about 28% anddownstream the check valve the mean turbulence intensity is about 10%. Thisindicates that the solution is in fact turbulent, especially considering the levelsare normalized with the maximum velocity. For turbulent flows like high-speedflow inside complex geometries like for example turbines and compressor theturbulence intensity is typically between 5% and 20% [34].

Figure 3.23 shows the volumetric flow rate across the check valve at a givenpressure difference. When comparing the flow rate between the three turbulentsolutions and the laminar solution the difference is small (about 4% betweenminimum and maximum) with the laminar solution showing the lowest flowrate for a given pressure difference.

The output parameter of interest for this study is pressure drop vs. flow rate.Since the steady state flow rate has been shown to be rather insensitive towhether the solution is turbulent or not, the choice is made to move forwardwithout a turbulence model, as the laminar solution requires less computationalresources.


(a) k − ω - SST. (b) k − ε - RNG.

(c) k − ε - Realizable .

Figure 3.22: Contours of turbulence intensity on center plane. Refer-ence velocity is Uref = 18 m/s measured at center of the jet exiting thegap.

0.05 MPa 0.15 MPa0

10

20

30

40

50

Exp.

Exp.

Flo

wra

te[l

/min

]

1 mm lift

w/o turbulence modelk-ω - SSTk-ε - RNGk-ε - Realizable

Figure 3.23: Volumetric flow rate at ∆p = 0.15 MPa and 1 mm lift.

3.3.5 Mesh study

As mentioned above in Section 3.3.2 the dynamic mesh will not include inflationlayers. The forces on the shim are mostly pressure induced and any frictioninduced forces are likely to be small in comparison. For this reason omittinginflation layers, and thus sacrificing resolution in the boundary layer, might notbe too much of a simplification. It does however make the resolution in the gap


much coarser at small openings and the velocity profile in the gap might not beaccurately resolved resulting in wrong flow rates for a given pressure drop.

Below follows a brief investigation of the impact mesh resolution and the ex-clusion of inflation layers has on the solution. Each mesh is studied in the fouroperation points stated in Table 3.5. Figure 3.24 shows the meshes at 0.1 mmlift. The meshes for the 1 mm lift case look like the one used for the turbulencemodel comparison, shown in Figure 3.17.

Table 3.5: Mesh study cases.

Lift height [mm] Pressure difference [MPa]Mesh study case 1: 0.1 0.05Mesh study case 2: 0.1 0.10Mesh study case 3: 1.0 0.05Mesh study case 4: 1.0 0.15

(a) Cell size: 0.2 mm. Withoutinflation. 293,000 cells.

(b) Cell size: 0.2 mm. With 5inflation layers. 312,000 cells.

(c) Cell size: 0.1 mm. Withoutinflation. 824,000 cells.

(d) Cell size: 0.1 mm. With 5inflation layers. 861,000 cells.

Figure 3.24: Meshes used in mesh study at 0.1 mm lift height.

The resulting volumetric flow rates, shown in Figures 3.25 and 3.26 show howat a small lift heights of the exclusion of inflation layers make a rather largedifference to the solution. At 0.1 mm lift the exclusion of inflation layers makesfor an increase in flow rate of 30-40% at the same pressure difference with thesemesh resolutions. The impact of removing the inflation layers has a greater


impact on the coarser of the two tetrahedral meshes. When comparing cellsize of the tetrahedral cells the difference in flow rate is minimal when inflationlayers are included. For the inflation-less meshes the difference in flow ratewhen going from 0.1 mm to 0.2 mm cell size is 5% and 7% for 0.05 MPa and0.10 MPa pressure difference respectively with the finer meshes as references.It makes sense that excluding inflation layers makes a substantial difference insolution as this small gap otherwise fits only one cell vertically with these cellsizes.

0.05 MPa 0.10 MPa0

2

4

6

8

Exp.

Exp.

Flo

wra

te[l

/m

in]

0.1 mm lift

0.2 mm tet. with infl. (312k elem.)


0.2 mm tet. w/o infl. (293k elem.)


Figure 3.25: Volumetric flow rate at set pressure difference and 0.1 mmlift.

0.05 MPa 0.15 MPa0

10

20

30

40

50

Exp.

Exp.

Flo

wra

te[l

/min

]

1 mm lift





Figure 3.26: Volumetric flow rate at set pressure difference and 1 mmlift.

The differences in solution due to mesh resolution is much smaller in the fullyopen state. Here there is no trend showing. On the finer mesh, excluding theinflation layers leads to an increase in flow rate of 2% at ∆p = 0.05 MPa and1% at ∆p = 0.15 MPa. For the coarser mesh the exclusion of inflation layersleads to a decrease in flow rate of 2% at ∆p = 0.05 MPa. At ∆p = 0.15 MPaonly a minimal difference in flow rate between the inflation layer equipped meshand the all tetrahedral mesh is observed.

Experimental data from the flow bench measurements have been added to thebar charts for reference. These are values from the tests done on the checkvalve without coil spring. It should be noted that for the CFD case, the shim


is modelled as undeformed. It is likely that the shim exhibits some bendingdeformation in the experiments. At least in the 0.1 mm gap case.

When comparing the CFD results to flow bench data for the 0.1 mm lift case isseems that excluding inflation layers brings the solution closer to the experimen-tal data. Without inflation layers and with these cell sizes the gap is only largeenough to fit a single row of cells. Including inflation layers greatly increasesthe resolution in the gap, and this should in theory make for a more accuratesolution. The fact that the inflation-less solutions correlate better to the flowbench results should not be interpreted as excluding inflation layers producesa more accurate solution. Since the fixed geometry CFD cases used for thiscomparison does not take the bending of the shim into account, the flow rate isunder predicted and cannot be directly compared to flow bench experiments.

The above comparisons have been made on fixed geometry cases. In the coupledcase mesh resolution might have stronger influence as the mesh resolution on theshim boundary affects the load resolution on the FSI interface. A comparison ismade on a steady state coupled case using the two tetrahedral meshes withoutinflation layers. These results are presented in section 4.2.

3.3.6 Boundary conditions

For the steady state calculations the flow is driven by a pressure differencebetween inlet and outlet. For model validation the same steady state pressuredrop vs. flow rate diagram produced in the flow bench is reproduced with theFSI model. Initially the mass flow rate was set at the inlet and static pressureset on the outlet. These boundary conditions proved too stiff. Major pressurespikes caused large displacements of the shim leading to negative cells in thedynamic mesh. A more robust setup is with a pressure inlet and pressure outlet.This way the p−Q diagram could be reproduced in the opposite way comparedto flow bench tests where pressure drop is measured output and flow rate isinput.

For the transient simulations where the aim is to reproduce a damper dy-namometer test a mass flow rate inlet can successfully be used without gettingthe large pressure peaks seen in the steady state case. The outlet is a pressureoutlet with static pressure set to zero as in the steady state setup. The pressurelevel is that of the gas pressure, which in the dynamometer tests are 0.6 MPa,is set under Operating Conditions in Fluent. The dynamometer test to be usedfor validation is the case of a 50 Hz sinusoidal piston velocity with a maximumamplitude of 0.25 m/s. This is represented at the inlet by a transient mass flowrate defined in a UDF (Appendix A). Figure 3.27 shows the mass flow rate atthe inlet as a function of time. In case there is a lag in the closing phase of thecheck valve the simulation time is extended a little bit beyond half the periodtime to capture the entire closing phase including possible shim bounce. As thesimulation is started there is a short delay before engaging the sinusoidal BCfunction. This serves as a check that the case is set up properly. If forces ordisplacements show big changes in value during these initial time step there islikely something wrong with the setup. During this initial phase the mass flowrate is set to a small positive value.

3.4. STRUCTURAL SETUP 39

0 0.5 1 1.5

·10−2

−2.0

0.0

2.0

·10−2

Sim.endtime

Time [s]

Mass

flow

rate

[kg/

s]

0 0.5 1 1.5

·10−2

−2.0

0.0

2.0

·10−2

Sim.endtime

Time [s]

Mas

sflow

rate

[kg/

s]

Figure 3.27: Transient inlet boundary condition.

3.3.7 Fluid properties

The damper fluid is modelled in Fluent as a compressible liquid using the Taitequation of state relating density to pressure under isothermal conditions [25,p. 417], (

ρ

ρ0

)n

=K0 + n(p− p0)

K0(3.17)

where ρ is density, p is absolute pressure, K is bulk modulus and n is densityexponent. Index 0 denotes reference values at reference pressure p0.

This helps reduce unphysical pressure spikes that can otherwise occur in calcu-lations with a moving mesh. Table 3.6 presents the material data for the fluid,and the inputs needed for the compressible liquid model in Fluent.

Table 3.6: Material data for the damper fluid.

Reference pressure [Pa]: 101325Reference density [kg/m3]: 843Reference bulk modulus [Pa]: 1.5·109

Density exponent [-]: 1Dynamic viscosity (constant) [kg/m·s]: 0.017

3.4 Structural setup

The structural domain need only include the FSI interface and any surface thatmight come in contact with it. Figure 3.28 shows the structural domain. Likethe fluid domain the structural domain is simplified by including only a 45◦

slice of the real geometry. The coil spring is modelled with a spring elementwhich is connected to the entire bottom surface of the shim as a deformablecontact, similar to a pressure load. The symmetry boundaries of the shim areconstrained in their planes by frictionless contacts.


Fixed support

Frictionless support

Fixed support

Spring element

Figure 3.28: Structural setup for the simplified case.

In the study by Leventhal [5] mentioned earlier the same spring element sim-plification was made. He was able to successfully capture both transient effectsat check valve opening and valve bounce at check valve closure. The differenceto this case is Leventhal’s check valve was of ball-in-seat type while the checkvalve of this study is of shim-against-seat type. The shim has some free playradially and is thus also allowed to pitch. As a coil spring is not perfectly axialin its reaction forces it is likely that the shim will have some radial and/or pitchdisplacement. These degrees of freedom are lost when making this symmetrysimplification to the domain.

The contact between shim and seat and, in the open state, contact betweenshim and spring collar are modelled with frictionless contacts with offsets to notpinch off the fluid mesh. Details of the contact detection were set after runningstandalone Structural cases with load representative of the fluid induced loads.Values on contact detection parameters were set in a trial-and-error approachto obtain convergence and are left to appendix (Appendix B). When runningstandalone Structural cases the spring element must have a non-zero internaldamping for the solution to converge. In the coupled cases no internal dampingis used as it is assumed small in comparison to the damping of the surroundingviscous fluid.

3.5 System Coupling setup

The procedure in System Coupling for coupling steady state cases and tran-sients cases are similar. In the transient case multiple coupling iterations arecarried out during each coupling step (time step). In the steady state case twoapproaches are possible - multiple explicit coupling steps or a single implicit cou-pling step. The steady state result will be the same regardless of the solutionprocedure as long as all quantities are converged.

3.5. SYSTEM COUPLING SETUP 41

3.5.1 Steady state

Both implicit and explicit steady solution procedures have been attempted onthis check valve case. The implicit approach with one coupling step and severalcoupling iterations in conjunction with force ramping proved to give the bestcontrol over displacement per coupling iteration. Too much displacement percoupling iteration lead to negative cells in the dynamic fluid mesh. Using forceramping provided some control over the displacement per time step. The fluidforces from Fluent to Structural are then linearly ramped up over the number ofminimum coupling iterations. When reproducing the pressure vs. flow rate di-agram from the flow bench measurements the steady state simulation is startedfrom a closed check valve. As the first solution is converged the boundary con-dition is changed and the solution procedure for the second operating point isstarted from previous solution. Convergence is confirmed by monitoring forceon and displacement of the shim as well as the force and displacement residuals.When solving the steady state case in 8 operating points starting from a closedcheck valve and increasing the pressure at inlet after each converged step untilthe shim was fully open a minimum of 10 coupling steps was enough to makesure the pressure was ramped up slowly enough to not cause any violent dis-placements. The maximum number of iterations was set to 13 to let the solutioniterate a few times with full loads applied. This was enough to converge thesolution in each operating point.

3.5.2 Transient

The transient simulations replicate a damper dynamometer test with a 50 Hzsinusoidal piston velocity. Maximum amplitude is 0.25 m/s. This experimentshow oscillations in pressure drop across the check valve with a frequency ofabout 670 Hz. The ambition is to see if these pressure oscillations occur also inthe FSI solution.

Initially a time step of 1·10−4 s is used. That is 20 time steps per the oscillationperiod seen in experiments. This time step proved to long as the coupling stepsdid not converge even with a as much as 10 Fluent iterations per coupling itera-tion and 8 coupling iterations per coupling step, resulting in 80 Fluent iterationsper time step. For a transient standalone Fluent simulation the ideal numberof iterations is somewhere between 5-10 iterations per time step according tothe Fluent Manual [24, p. 1470]. If more iterations are needed to converge eachtime step, the time step is too large and it is more efficient to reduce the timestep than adding to the number of iterations. For an FSI case this rule of thumbdoes not directly apply as the geometry will change within the time step andmore iterations will be needed than for a standalone case but it serves at leastas a guideline indicating that the time step is to large.

With a shorter time step of 2.5·10−5 and fewer Fluent iterations and couplingiterations the solution converged better in the first few time step. As the shimstarts moving the solution grows unstable within the time step. These instabil-ities are counteracted using Solution Stabilization in Fluent which slows downthe pressure response at the shim boundary.


The scale factor for the Solution Stabilization is found in a trial-and-error ap-proach starting with no stabilization and increasing the scale factor until thesolution is stable. Note that stable in this context refers to the stability withineach time step. Figure 3.29 shows integrated pressure on the shim and dis-placement of the shim with different scale factor values. Note that for the caseof Solution Stabilization scale factor = 0.008 the initial condition is slightlydifferent (lower initial mass flow rate).

0 200 4000.000

0.500

1.000

1.500

2.000

Iteration

Forc

e[N

]

(a) Force.

0 200 4004.120

4.122

4.124

4.126

4.128·10−3

IterationD

isp

lace

men

t[m

]

(b) Displacement.

0 200 4000.000

2.000

4.000

6.000·104

Iteration

∆p

[Pa]

(c) Pressure drop.

SS Scale factor = 0.000SS Scale factor = 0.005SS Scale factor = 0.008SS Scale factor = 0.010

Figure 3.29: Tuning of Solution Stabilization scale factor. The plotsshow the convergence of solution quantities within each time step duringthe first time steps of the transient simulation. Iterations on the horizon-tal axis refers to Fluent iterations. The difference in magnitude for thecase of SS Scale factor = 0.008 is due to a difference in initial conditions.

A scale factor of 0.008 is enough to stabilize the solution. Seen in Figure 3.29 ishow displacements converge nicely but the shim force does not fully converge inthese first time steps. As for the data transfer residuals, the normalized RMSresiduals for the loads sent from Fluent and displacements sent from Structuralare below 1 · 10−2 for the major part of the simulation. Figure 3.30 shows theresiduals for the load transfer from Fluent and displacement from Structural.

Another aspect to consider when deciding on time step is stability for the fluidsolution. The time stepping in Fluent is implicit and there is no stability cri-

3.6. FINAL TRANSIENT MODEL 43

Figure 3.30: Data transfer residuals.

terion to be met when deciding time step but for an efficient solution it isrecommended in Fluent that the Courant number does not exceed 20-40 in sen-sitive transient regions [25, p. 1471]. The transient cases studied in this M.Sc.degree project involve maximum fluid velocities of about 9 m/s s in the shim-seat gap, which with a minimum cell size of 2·10−4 m in the region of highestflow velocities translates to a Courant number of about

CFL =u∆t

∆x=

9 · 2.5 · 10−5

2 · 10−4= 2.25 (3.18)

which is well within the recommended limit.

3.6 Final transient model

In the final stage of the project the 45◦ simplification is dropped and a simu-lation is carried out on a larger domain. As the transient simulations are tobe validated with dynamometer test the final CFD domain geometry is basedon the dynamometer damper end piece. Figure 3.7 shows the real geometry ofthe damper and Figure 3.31 shows the geometry used for the fluid domain inthe final transient simulation. The hole in the center of the check valve hasbeen closed in the fluid domain. This is a stagnant region and the effect on thesolution is assumed negligible.

Most of the settings used in the simplified case are carried over to this largercase with the difference that the symmetry boundary conditions are no longerpresent. The mesh is larger (1.3·106 cells) but the cell sizes remain the same.The scale factor for the Solution Stabilization in Fluent is reduced to 0.002without any instabilities showing in the solution. It might be possible to reducethe scale factor even more but because of the long computation times for thiscase these settings are not explored in-depth.

A possible explanation to the higher stability of this larger case compared to thesimplified case might be the less restrictive constraints on shim displacement.In the simplified model the shim is constrained to the two symmetry planes,


Inlet

Outlet

(a) Entire domain.

Swept hex.mesh

Deformingmesh zone

Unstructuredtet. mesh

(b) Mesh interior.

Figure 3.31: Fluid mesh for the final transient simulations.

while for the larger model the shim is free to move asymmetrically. As the shimis allowed to tilt when lifting of the seat as opposed to lifting straight down inan axisymmetric translation, the initial lift-off off the shim might be less abruptin the larger domain. The stability issues might also originate in the symmetryconditions in the CFD-domain, as this is the only major difference in the fluidmodel between the simplified and the final model. Aside from the size of thedomain that is.

In the structural model the spring is still modelled by a spring element, andsince the symmetry boundary condition is dropped the shim is now allowedsome radial motion and is also allowed to pitch. Radial motion is constrainedby frictionless contacts on the inner boundary with a small offset to not pinch offthe fluid mesh. Figure 3.32 shows the solid geometry and boundary conditions.

Spring load appliedunder the shim.

Springelement

Fixed supports ontop, bottom andinnermost boundaries.

Frictionless contacts onouter cylindrical boundary,seat and stop.

Figure 3.32: Solid domain (section view).

CHAPTER4Results

Below follows the results from experiment and simulation. The steady stateFSI results are compared to the quasi-steady state results from the flow benchtests. These results are presented in the form of p −Q diagram as is standardin shock absorber analysis. The transient FSI results are compared to thedynamometer tests. These results are presented as time series of pressure dropand displacement, as well as solution contours on a center plane.

4.1 Flow bench measurements

The results of the flow bench measurements are presented in p − Q diagrams.Figure 4.1 shows the results from the flow bench measurements without thespring installed for different lift heights. These results themselves are not veryinteresting but they serve as a validation tool for the CFD model, since in thistest the lift height is known and can thus be replicated in a fixed geometry CFDcase. The rings mark the values used for comparison in the mesh study.

Figure 4.2 and 4.3 show the results from the measurements with the springinstalled for the two different maximum lift height. Initially the pressure dropincreases rapidly with flow rate until the pressure build-up is large enough toovercome the spring preload, whereupon the pressure drop increases smoothlywith flow rate. In the 1 mm maximum lift case the 3 N/mm spring setup p−Qcurve intersects with the no-spring setup at around 30 l/min. This is where thepreloaded shim reaches maximum lift. With the stiffer spring the shim reachesmaximum lift at a flow rate outside of the test range.

Comparing the setups with different maximum lift height (1 mm vs. 2 mm) it isinteresting to note that the pressure drop is lower for the setup with the highermaximum lift even before the shim has reached maximum lift. From about 20l/min and above there is a noticeable difference between the two setups withdifferent max. lift. Perhaps the shim opens in an asymmetrical fashion bypitching and hitting the 1 mm stop first on one side and then continue to openuntil the shim is in full contact with the stop. The flow bench is closed and it

46 CHAPTER 4. RESULTS

0 10 20 30 400.00

0.05

0.10

0.15

0.20

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a]

0.1 mm lift0.4 mm lift0.7 mm lift1.0 mm lift1.4 mm lift2.0 mm lift

Figure 4.1: Pressure drop at constant lift height, measured in flowbench.

0 10 20 30 400.00

0.05

0.10

0.15

0.20

Max. lift

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a]

Max lift: 1 mm

w/o spring



Figure 4.2: Pressure drop at 1 mm maximum lift, measured in flowbench..

it not possible to visually monitor the shim displacement and the steady statemodel is axisymmetric and thus limited to axial translation and axisymmetricalbending. The final transient model does not have these constraints and thoseresults show how the pressure load is not axisymmetric and how the shim opensby both translating and pitching. This is likely what happens in flow bench aswell, as the inlet is coming from the side like in the dynamometer damper.

4.2. STEADY STATE FSI 47

0 10 20 30 400.00

0.05

0.10

0.15

0.20

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a]

Max lift: 2 mm

w/o spring



Figure 4.3: Pressure drop at 2 mm maximum lift, measured in flowbench..

4.2 Steady state FSI

Figure 4.4 shows the the steady state FSI solutions for a number of operatingpoints, computed on the simplified domain, together with flow bench data. Forthe flow bench measurements the upstream pressure is taken in the large cavityholding the check valve fixture, where the flow is assumed stagnant. The CFDmodel omits this cavity and instead the upstream pressure is taken at the inlet tothe fixture. To have comparable data the pressure drop from the FSI simulationsis the difference in total pressure at inlet and static pressure at outlet.

∆pfsi = pin0 − pout (4.1)

The solution in some of the operating points have been found on two differentmeshes - 0.1 mm cells and 0.2 mm cells. Both meshes are unstructured tetrahe-dral meshes with uniform resolution and without inflation layers. The differenceis minimal.

For the case without spring where the check valve is fully open for all flow ratesthe FSI model underestimates the pressure drop with 10%-20%. For the twosetups with the coil spring installed the model underestimates the pressure dropat flow rates over 10 l/min and below that slightly overestimates the pressuredrop.

Referring to Figure 4.4, one solution point for the 8 N/mm setup stands out(marked ”Discrepancy”). At Q = 33 l/min the difference to the flow benchdata is 23%. At this flow rate the check valve is fully open. When solving forthe steady state solution the pressure load from the fluid field is ramped over anumber of coupling iterations. During the solve process the shim slowly movedin small increments with each coupling iteration as the pressure was linearlyramped up. When the shim got close to the stop the it suddenly moved in alarge increment to a fully open state. A possible explanation is that the flow in


0 5 10 15 20 25 30 35 40 450.00

0.05

0.10

0.15

0.20

Discrepancy

Flow rate [l/min]

Pre

ssu

red

rop

[MP

a]

Flow bench, w/o spring Flow bench, 3 N/mm

Flow bench, 8 N/mm FSI, w/o spring, 0.2 mm tet.

FSI, 3 N/mm, 0.2 mm tet. FSI, 8 N/mm, 0.2 mm tet.

FSI, 3 N/mm, 0.1 mm tet. FSI, 8 N/mm, 0.1 mm tet.

Figure 4.4: Steady state pressure drop across the check valve vs. flowrate. A comparison between computed results from the simplified FSImodel and flow bench measurements for three check valve setups. Max-imum lift is 1 mm, preload for the softer 3 N/mm spring is 3.9 N andpreload of the stiffer 8 N/mm spring is 10.4 N.

the gap formed between shim and stop generates a low pressure region pullingthe shim fully open. Between the low pressure side of the shim and the shimstop there is a small offset (20 µm) to not pinch of the fluid mesh.

The steeper pressure-flow rate curve in the ”nose” region (up to Q = 2 l/minfor the 8 N/mm setup) for the simulation compared to flow bench measurementmight be caused by the symmetry condition in the structural model. Seen inthe transient simulation is how the shim opens not by pure axial translation butinstead by pitching. The same is likely to happen in the flow bench. This mightexplain why the pressure build up is not as sudden as in the FSI simulationwhere the shim opens in pure axial translation.

4.3 Dynamometer tests

Figure 4.5 shows the pressure difference over the check valve measured on thedynamometer damper, as function of time. The three tested check valve setupsshow similar behaviour - an initial peak in pressure followed be oscillations ataround 670 Hz. What is interesting is that the oscillation period is the samefor the different spring rates, indicating that the oscillations are not effects

4.4. TRANSIENT FSI 49

attributed to the coil spring.

1 1.2 1.4 1.6 1.8 2

·10−2

0.00

0.05

0.10

0.15

T ≈ 1.5 · 10−3 s

Time [s]

Pre

ssu

red

rop

[MP

a]




Figure 4.5: Dynamometer results. 50 Hz sinusoidal piston velocity withmaximum amplitude 0.25 m/s. ∆p vs. time in rebound stroke.

4.4 Transient FSI

The same setups tested in the dynamometer are also simulated in the FSI model.The results from the simplified model are presented in Figure 4.6. The pressuredrop is highest as the check valve first opens. The pressure drop then oscillatesbefore settling and then decreasing for the rest of the duration. As in theexperiments the oscillation period is the same for the softer and the stifferspring setup. The frequency of these oscillations are about three times as highas those seen in the experiments.

0 0.2 0.4 0.6 0.8 1

·10−2

0.00

0.05

0.10

T ≈ 0.5 · 10−3 s

Time [s]

Pre

ssu

red

rop

[MP

a]




Figure 4.6: FSI results from the simplified model. ∆t = 2.5 · 10−5 s.

The larger model is tested only on one of the check valve setups - the softerspring with the lower preload. Figure 4.7 shows pressure drop vs. time from thesimplified and the larger model as well as the experimental data. The results


from the larger model show a higher pressure peak initially. The followingoscillations are slightly higher in frequency than those seen in the smaller model.The biggest difference to the smaller model is the behaviour at the end of thecycle. The results from the larger model shows a negative total pressure dropat the end of the cycle.

0 0.2 0.4 0.6 0.8 1

·10−2

0.00

0.05

0.10

Time [s]

Pre

ssu

red

rop

[MP

a]

Simpl. 45◦ domainFull 360◦ domainDynamometer

Figure 4.7: FSI results from the simplified and the final larger model.Spring rate: 3 N/mm, preload: 3.9 N, ∆t = 2.5 · 10−5 s.

Figure 4.8 shows the axial displacement of the shim as function of time. Dis-placement is measured on the nodes shown in Figure 4.9. The smaller model isconstrained to axisymmetric displacement, while in the larger model the shim isfree to move in all spatial dimensions. For the larger model the displacement ismostly a pitch displacement. The shim opens the most on the side of the inlet.

0 0.2 0.4 0.6 0.8 1

·10−2

−1.0

−0.5

0.0

·10−3

Time [s]

Axia

ld

isp

lace

men

t[m

] Simpl. 45◦ domain, node 1Simpl. 45◦ domain, node 2Full 360◦ domain, node 1Full 360◦ domain, node 2Full 360◦ domain, node 3

Figure 4.8: FSI results from final model. Spring rate: 3 N/mm, preload:3.9 N, ∆t = 2.5 · 10−5 s. Shim displacement vs. time.

The largest displacement is seen at t = 0.6 ·10−2 s. Figure 4.10 shows the staticpressure on a center plane at this solution time. The inlet is coming from theright in the figure. Figure 4.11 shows the velocity magnitude in the same plane.Seen in both figures is how shim lifts only so slightly on the side farthest from

4.4. TRANSIENT FSI 51

(a) Simplifieddomain. (b) Full domain.

Figure 4.9: Position of displacement monitors.

the inlet (to the left in the figures), and lifts almost to the stop at the inlet side.

Figure 4.10: Pressure distribution on center plane at t = 0.6 · 10−2 s(final model).

Figure 4.11: Velocity distribution on center plane at t = 0.6 · 10−2 s(final model).

CHAPTER5Conclusion and Discussion

Both steady state and transient simulations have been carried out and com-pared to experiments. The models, both steady state and transient, have beensimplified to isolate the check valve and to make it possible to solve the caseson a laptop computer. These simplifications include:

• Reduced fluid domain.

• Isothermal compressibility model.

• No turbulence model.

• Coil spring modelled using a spring element.

• No inflation layers in the deforming mesh zone.

5.1 Conclusion

The steady state simulations show satisfactory correlation to flow bench mea-surements at low flow rates. As flow rate increases so does the error. It shouldbe noted that the CFD domain does not include the entire length of pipingleading up to the downstream sensor in the flow bench, but is based on thegeometry of the check valve fixture alone. There is likely an additional pressuredrop between the outlet of the fixture and the position of the downstream pres-sure sensor. This could explain why the FSI model predicts a lower pressuredrop than seen in the flow bench at the same flow rate.

The transient FSI model show poor correlation to dynamometer experiment.Both the transient behaviour in the beginning of the cycle (oscillations in pres-sure) and the amplitude of the pressure drop as the oscillations settled. This isfor both the simplified model and the final larger model. The pressure peak hada lower amplitude in the model and the oscillations seen in the model had threetimes shorter period time than those seen in the experiment. Here it is impor-tant to note that the sampling frequency in the dynamometer experiment were

54 CHAPTER 5. CONCLUSION AND DISCUSSION

to low to resolve those frequencies. When considering the level of simplificationsmade to the models, a better correlation was not expected.

It should also be noted that the dynamic effects seen in the dynamometer exper-iments not necessarily have their origin in the check valve. In the FSI model thecheck valve can easily be isolated while this is not the case in the experimentalsetup. Structural compliance in the damper and friction in bushings might affectthe results. The boundary condition in CFD domain was a sinusoidal mass flowrate to replicate the sinusoidal velocity of the dynamometer. However, look-ing at the velocity profile of the dynamometer (Figure 3.8b) the profile is notperfectly sinusoidal. Saying something about the accuracy of the transient FSImodel would require a larger domain including more of the damper geometryor perhaps an experimental setup that better isolates the check valve.

The CFD mesh does not contain inflation layers on the wall boundaries in thedeforming mesh region. This to allow the shim to close to a minimal gap. Ithas been shown how this simplification has an effect on the solution at smalllifts (0.1 mm), 30%-40% higher flow rate at a given pressure. For large lifts (1.0mm) the exclusion of inflation layers had minimal effect.

For this application the choice of turbulence model has shown to be non-criticalfor steady state simulations. Two of the most commonly used turbulence mod-els have been compared to each other and to the solution without turbulencemodelling. In the flow field downstream there is a noticeable difference in howquickly the jet dissipates. For the solution without turbulence model the jetextends the furthest. However, this does only slightly affect the steady statepressure difference across the check valve.

With steady state results being insensitive to turbulence modelling the choicewas made to move on to transient simulations without a turbulence model.Although this simplification was motivated for the steady state cases the samesimplification might not be valid for transient cases, as this might lead to toolow levels of internal damping in the fluid and thus affecting dynamic behaviourof the check valve. A turbulence model might increase the damping in thefluid (via viscous friction losses in the modelled Reynolds stresses). For furthertransient studies it is recommended to study the impact of turbulence modellingnot only on the steady state solution but the impact on transient as well.

5.2 Discussion

The goal of this M.Sc. degree project was to set up and, above all, validate anFSI model. In doing so a number of simplifications have been made. Perhaps toomany for any model-experiment comparison to be relevant. For the steady statevalidations, the CFD domain only included the fixture for mounting the checkvalve in the flow bench. It would perhaps have been a better idea to includethe full length of piping in the flow bench leading up to the pressure sensorsfor a better comparison. It should also be noted that the downstream pressuresensor in the flow bench is not mounted flush with the boundary. Instead it isconnected via a junction. There might be disturbances across the junction caus-ing significant dynamic pressures. How much this affects the pressure reading is

5.2. DISCUSSION 55

unclear. This might explain the difference in model and experiment. The flowbench is typically used for relative measurements to compare different designsand settings, and thus is has not been designed for absolute measurements.

Because the flow bench is run in reverse to its default setup the fixture ismounted on top of an adapter plate that reroutes inlet and outlet piping leadingin to the flow bench main cavity. This obstruction to the flow, albeit small, be-tween the fixture outlet and the downstream pressure sensor may give rise to ameasurable pressure drop in addition to the pressure drop over the check valve.This might also be an explanation to the poor steady state model-experimentcorrelation at high flow rates.

With steady steady state results being insensitive to turbulence modelling thechoice was made to move on to transient simulations without a turbulencemodel. Although this simplification was motivated for the steady state casesthe same simplification might not be valid for transient cases, as this might leadto too low levels of internal damping in the fluid and thus affecting dynamicbehaviour of the check valve. A turbulence model might increase the dampingin the fluid (via viscous friction losses in the modelled Reynolds stresses). Forfurther transient studies it is recommended to study the impact of turbulencemodelling not only on the steady state solution but the impact on transient aswell.

The reduction of the domain to a 45◦ slice with symmetry boundary conditionswas done to be able to solve the cases at a laptop computer. The results fromthe final model, which does not have the same restrictions in degrees of free-dom, show how the shim displacement is highly asymmetrical. The 45◦ slicesimplification is therefore not recommended for further studies. It did howeverprove valuable to have made this simplification when exploring dynamic meshsettings and debugging the model.

In this study the steady state FSI was a step in validating the transient modelto come. If steady state results are the goal the better practice would be to usethe dynamic mesh described in Section 3.3.2, which includes inflation layers butprevents the shim from being able to close to a very small gap, as it would notbe necessary to start from a very small gap when just the steady state solutionis of interest.

It was assumed early on that the simulation had to be started from a very smallgap in order to capture the pressure peak at the start of the cycle. Perhaps thetransient simulation can be started from a larger gap without losing out on anytransient effects (oscillations in pressure). It would be a good idea for futuretransient studies to do investigate whether starting from a larger initial gapwould affect the solution. Starting from a larger gap would allow for inflationlayers also in the deforming zone, making for better accuracy. A 20 µm initialgap was chosen as this was done by Leventhal [5] in an other check valve study.However, the maximum lift on his check valve was smaller. In this case a largergap might work fine. An approach that was suggested early on was to usea porous media to block the flow and as soon as the shim moves, remove theporous media. This would allow for the same pressure build-up as when startingfrom a very small gap.

The spring element force was applied on the shim as a distributed load on the

56 CHAPTER 5. CONCLUSION AND DISCUSSION

entire lower boundary of the shim. A more accurate representation would beto project the upper contact surface of the coil spring onto the shim and applythe force on that area, or better yet include the coil spring in the model asan elastic body and drop the spring element simplification altogether. Becauseof this simplified boundary condition the stresses in the shim are likely to befar from representative of the real case, and have therefore been left out of thereport.

5.2.1 2-way FSI as a development tool

FSI can most definitely be a useful tool in shock absorber development. Thesteady state model showed good correlation to experiments around openingpressure, and can therefore be used with confidence when evaluating new designconcepts as to whether the design will work or not (open or remain closed). Allthe steady state simulations in this M.Sc. degree project have been run on alaptop computer (4 cores). This was not especially limiting as solving for oneoperating point (on the smaller domain) was done in approximately 15 minutes.

Transient FSI studies require quite a lot more work compared to a steady statecase, as there is also the need to ensure convergence in every time step. Thecomputational requirements are typically higher for a transient simulation com-pared to steady state. This holds true for standalone CFD cases as well asstandalone FEM cases. When coupling the two the increase in computationalrequirements become even more pronounced. The transient simulations on thesmaller domain could be solved overnight (≈ 14 hours) on 4 cores. The finaltransient simulation was solved on a 62-core cluster (Fluent using 58 cores andstructural using 4 cores) with the computation time being 4 days.

To summarize, steady state FSI is easy to set up and require small computationalresources. Transient FSI requires more work and more computational resources.

5.2.2 Suggestions for further studies

For further FSI analysis of this check valve it is recommended to use the fullgeometry as computational domain. The simplified 45◦ domain was useful fordebugging and exploring dynamic mesh setting but the results differed from theones produced by the final larger model. To improve the structural model, thecoil spring could be included as an elastic body. It is however not recommendedto include the coil spring in the fluid domain. Doing so will probably make thedynamic mesh setup troublesome. Instead a dummy body of simpler geometrycould be included in the fluid domain as a representation of the coil spring.

It is also recommended that a study on the impact of the initial gap is done be-fore taking the transient model further. Here it was assumed that the transientsimulation must be started from a very small gap in order to get an accuratepressure build-up at the beginning of the cycle. If this is not the case, andthe simulation can be started from a larger gap it is recommended to go withdynamic mesh approach #4 described in Section 3.3.2, which included inflationlayers in the deforming zone.

5.2. DISCUSSION 57

For steady state simulations the simplification of not using a turbulence model toreduce computation time has been motivated. For transient simulation however,it is recommended to use a turbulence model because of the effects it might haveon internal damping in the fluid.

While the simplified domain was meshed with a sweep method everywhere ap-plicable the mesh of the larger model is slightly simpler in its setup, with onlythe cylindrical parts of inlet and outlet being swept. Sweepable regions can befound also in other parts of the domain. If moving forward with this model itcould be a good idea to spend some time on this mesh to reduce the number ofelements by identifying more regions that can be swept.

For this M.Sc. degree project the pressure drop across the check valve has beenused as the solution quantity for validation. It would be a good idea to lookalso at displacement. In the start of this project the ambition was to plan andexecute a dynamometer test with a transparent cavity for the check valve. Shimdisplacement can then be visually monitored, and captured using a high speedcamera. This might be a suitable topic for an other degree project.

References

[1] G. E. Moore. ”Cramming more components onto integrated circuits”. In:Electronics 38.8 (1965), pp. 114–117.

[2] N.-G. Nygren. Inside TTX - The Ohlins TTX40 Manual. 1st ed. Sweden:Ohlins Racing AB, May 2005.

[3] Sandvik 20C for shock absorber shims. Datasheet. 2013. url: http://www.smt.sandvik.com/en/materials-center/material-datasheets/

strip-steel/sandvik-20c-for-shock-absorber-shims/ (visited on10/13/2014).

[4] D. Borglund and D Eller. Aeroelasticity of Slender Wing Structures inLow-speed Airflow : Lecture Notes. Stockholm, Sweden: Royal Institute ofTechnology, 2013).

[5] L. Leventhal. CAE Applications in hydraulics - Experimental and analyt-ical study of a check-valve. Tech. rep. SAE Technical Paper 2003-01-1605.2003. doi: 10.4271/2003-01-1605.

[6] ADINA R&D, Inc. Offical webpage. url: http://www.adina.com (visitedon 10/13/2014).

[7] S. M. Price, D. R. Smith, and J. D. Tison. ”The effects of valve dy-namics on reciprocating pump reliability”. In: Proceedings of the TwelfthInternational Pump Users Symposium. Ed. by J. C. Bailey. Texas A&MUniversity, Mar. 1995, pp. 221–230.

[8] Exa PowerFlow. Official webpage. url: http://www.exa.com/powerflow(visited on 11/03/2014).

[9] XFlow. Official webpage. url: http://www.xflowcfd.com (visited on11/03/2014).

[10] C. L. Fefferman. ”Existence and Smoothness of the Navier-Stokes Equa-tion”. In: The Millennium Prize Problems. Ed. by J. A. Carlson, A. Jaffe,and A. Wiles. Providence, Rhode Island: American Mathematical Society,2006, pp. 57–67. isbn: 0-8218-3679-X.

[11] Y. Du. ”Numerical simulation of mechanical and thermal fluid-structureinteraction in labyrinth seals”. PhD thesis. TU Darmstadt, Aug. 2010.url: http://tuprints.ulb.tu-darmstadt.de/2253/.

[12] P. K. Dowling, I. M. Kundu, and D. R. Cohen. Fluid Mechanics. FifthEdition. Boston: Academic Press, 2012. isbn: 978-0-12-382100-3.

60 REFERENCES

[13] COMSOL Multiphysics. Official webpage. url: http://www.comsol.se(visited on 10/13/2014).

[14] H. Zhang and K.-J. Bathe. ”Direct and iterative computing of fluid flowsfully coupled with structures”. In: Computational Fluid and Solid Me-chanics. Ed. by K. J Bathe. Vol. 1. Elsevier, June 2001, pp. 1440–1443.

[15] S. Piperno and C. Farhat. ”Design of efficient partitioned procedures forthe transient solution of aeroelastic problems”. In: Revue Europeenne desElements 9.6-7 (2000), pp. 655–680. doi: 10 . 1080 / 12506559 . 2000 .

10511480.

[16] C Michler et al. ”The relevance of conservation for stability and accu-racy of numerical methods for fluid-structure interaction”. In: ComputerMethods in Applied Mechanics and Engineering 192.37 (2003), pp. 4195–4215.

[17] A. Dervieux et al. ”Total energy conservation in ALE schemes for com-pressible flows”. In: European Journal of Computational Mechanics 19.4(2010), pp. 337–363.

[18] Fluid-Structure Interaction (FSI) with ANSYS Fluent 15.0. ANSYS Cus-tomer Portal. Training material, Lecture slides. May 2014.

[19] M. G. Doyle, S. Tavoularis, and Y. Bourgault. ”Application of parallelprocessing to the simulation of heart mechanics”. In: HPCS. Ed. by D. J.K. Mewhort et al. Vol. 5976. Springer, 2009, pp. 30–47. doi: 10.1007/978-3-642-12659-8_3.

[20] Y. Bazilevs, K. Takizawa, and T. E. Tezduyar. Computational Fluid-Structure Interactions: Methods and Application. Chichester, UK: Wiley,2013. isbn: 978-0-470-97877-1.

[21] J. E. Akin, T. E. Tezduyar, and M. Ungor. ”Computation of flow prob-lems with the Mixed Interface-Tracking/Interface-Capturing Technique(MITICT)”. In: Computers & Fluids 36.1 (2007), pp. 2–11. doi: 10.1016/j.compfluid.2005.07.008.

[22] ANSYS Fluent Dynamic Mesh (Moving Deforming Mesh) 14.5. ANSYSCustomer Portal. Training material, Lecture slides. Nov. 2013.

[23] ANSYS Meshing User’s Guide. 15.0. Canonsburg, Pennsylvania: ANSYS,Inc., Nov. 2013.

[24] ANSYS Fluent Theory Guide. 15.0. Canonsburg, Pennsylvania: ANSYS,Inc., Nov. 2013.

[25] ANSYS Fluent User’s Guide. 15.0. Canonsburg, Pennsylvania: ANSYS,Inc., Nov. 2013.

[26] B. Launder and D. Spalding. ”The numerical computation of turbulentflows”. In: Computer Methods in Applied Mechanics and Engineering 3.2(1974), pp. 269 –289. doi: 10.1016/0045-7825(74)90029-2.

[27] D. C. Wilcox. ”Formulation of the k-ω turbulence model revisited”. In:AIAA Journal 46.11 (2008), pp. 2823–2838.

[28] T.-H. Shih et al. ”A new k-ε eddy viscosity model for high Reynolds num-ber turbulent flows - Model development and validation”. In: Computers& Fluids 34.3 (1995), pp. 227–238. doi: 0.1016/0045-7930(94)00032-T.

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[29] S. A. Orszag et al. ”Renormalization Group Modeling and TurbulenceSimulations”. In: Near-wall turbuent flows. Ed. by R. M. C. So, C. G.Speziale, and B. E. Launder. Amsterdam, Netherlands: Elsevier, 1993,pp. 1031–1046.

[30] F. R. Menter. ”Two-equation eddy-viscosity turbulence models for engi-neering applications”. In: AIAA Journal 32.8 (1994), pp. 1598–1605. doi:10.2514/3.12149.

[31] H. K. Versteeg and W. Malalasekera. An Introduction to ComputationalFluid Dynamics: The Finite Volume Method. Second Edition. PearsonPublications, 2007. isbn: 978-81-317-2048-6.

[32] A. Hashiehbaf and G. P. Romano. ”Experimental investigation on circu-lar and non-circular synthetic jets issuing from sharp edge orifices”. In:Proceedings of 17th International Symposium on Applications of LaserTechniques to Fluid Mechanics. Lisbon, Portugal, July 2014.

[33] C. Bogey and C. Bailly. ”A study of the influence of the Reynolds numberon jet self-similarity using large-eddy simulation”. In: Direct and Large-Eddy Simulation VII. Ed. by V. Armenio, B. Geurts, and J. Frohlich.Vol. 13. Springer Netherlands, 2010, pp. 11–16. isbn: 978-90-481-3651-3.doi: 10.1007/978-90-481-3652-0_2.

[34] CFD Online - Turbulence intensity. Webpage. 2012. url: http://www.cfd-online.com/Wiki/Turbulence_intensity (visited on 10/13/2014).

APPENDIXAUDF’s and Profiles

”Dummy” UDF for inflation layers in separatecell zone

#include "udf.h"

DEFINE_GRID_MOTION(dummy_motion, domain, dt, time, dtime)

{

return;

}

Profile for previewing dynamic mesh motion

(

(open_n_close_valve point 4 1)

(time 0.0000 0.0095 0.0096 1.0000)

(v_y -0.1 -0.1 0.1 0.1)

)

64 APPENDIX A. UDF’S AND PROFILES

UDF for transient inlet BC (360◦ case)

#include "udf.h"

#define PI 3.14159265359

static real time_prev = 0.0;

static real mfr_in_0 = 0.0001; /* Initial flow rate [kg/s] */

static real mfr_in_max = 0.21452; /* Maximum flow rate [kg/s] */

static real freq = 50; /* Frequency [Hz] */

static real time_init = 0.0003; /* Delay to let flow field initialize [s] */

DEFINE_PROFILE(mfr_in_t,thread,pos)

{

#if !RP_HOST

face_t f;

real time, dtime, mfr_in_ampl, mfr_in;

time = CURRENT_TIME;

dtime = CURRENT_TIMESTEP;

mfr_in_ampl = mfr_in_max - mfr_in_0;

/* Update mass flow inlet at each time step */

if(fabs(time_prev-time) > 0.2*dtime)

{

/* Current mass flow inlet */

if(time <= time_init)

{

mfr_in = mfr_in_0;

}

else

{

mfr_in = mfr_in_ampl * sin(2*PI*freq * (time-time_init)) + mfr_in_0;

}

/* Loop over the inlet faces */

begin_f_loop(f,thread)

{

F_PROFILE(f,thread,pos) = mfr_in;

}

end_f_loop(f,thread)

/* print new mass flow rate to TUI*/

Message0("\n\n ************* MASS FLOW INLET INFO *************\n");

Message0("\n Mass flow rate : %5.3e kg/s", mfr_in);

Message0("\n Flow time : %5.3e s", time);

Message0("\n\n ************************************************\n\n");

/* assign current time */

time_prev = time;

}

#endif

}

APPENDIXBModel settings (360◦ case)

Fluent setup

Floating point format . . . . . Double precision

Solution Setup

SolverType . . . . . . . . . . . . . . . . . . . . . . . . Pressure-BasedTime . . . . . . . . . . . . . . . . . . . . . . . . Transient

ModelsEnergy. . . . . . . . . . . . . . . . . . . . . . . OffViscous Model . . . . . . . . . . . . . . Laminar

MaterialsFluid . . . . . . . . . . . . . . . . . . . . . . . . Damper fluid (refer to Table 3.6)

Boundary conditionsInlet . . . . . . . . . . . . . . . . . . . . . . . . mass-flow-inlet, (refer to Figure 3.27)Outlet . . . . . . . . . . . . . . . . . . . . . . pressure-outlet, 0 PaOperating Pressure . . . . . . . . . . 1·106 Pa

Dynamic MeshMesh Methods . . . . . . . . . . . . . . Smoothing, Remeshing

SmoothingMethod DiffusionDiffusion Function Boundary DistanceDiffusion Parameter 0

66 APPENDIX B. MODEL SETTINGS (360◦ CASE)

RemeshingRemeshing Methods Local Cell, Local FaceMin. Length Scale 6·10−5 mMax. Length Scale 2.5·10−4 mMax Cell Skewness 0.95Size Remeshing Interval 1

Solution StabilizationMethod . . . . . . . . . . . . . . . . . . . . . . . coefficient-basedScale Factor . . . . . . . . . . . . . . . . . . . 0.002

Solution

Pressure-Velocity CouplingScheme . . . . . . . . . . . . . . . . . . . . . Coupled

Spatial DiscretizationPressure . . . . . . . . . . . . . . . . . . . . . Second OrderDensity . . . . . . . . . . . . . . . . . . . . . Second Order UpwindMomentum . . . . . . . . . . . . . . . . . Second Order Upwind

Transient FormulationScheme . . . . . . . . . . . . . . . . . . . . . First Order Implicit

Solution ControlsFlow Courant Number . . . . . . 20Explicit Relaxation Factors . . Momentum: 0.75

Pressure: 0.75

Time StepMax Iterations/Time Step 4

Structural setup

Connections

Contacts:

DefinitionType . . . . . . . . . . . . . . . . . . . . . . . . Frictionless

AdvancedFormulation . . . . . . . . . . . . . . . . . Augmented LagrangeNormal Stiffness . . . . . . . . . . . . ManualNormal Stiffness Factor . . . . . 0.5Pinball Region . . . . . . . . . . . . . . RadiusPinball Radius . . . . . . . . . . . . . . 2.5·10−3 mTime Step Controls . . . . . . . . . None

APPENDIX B. MODEL SETTINGS (360◦ CASE) 67

Geometric ModificationInterface Treatment . . . . . . . . . Add Offset, No RampingOffset . . . . . . . . . . . . . . . . . . . . . . . 2.0·10−5 m (seat and stop),

1.0·10−4 m (inner boundary)

Spring element:

DefinitionType . . . . . . . . . . . . . . . . . . . . . . . . LongitudinalLongitudinal Stiffness . . . . . . . 3000 N/mPreload . . . . . . . . . . . . . . . . . . . . . Free LengthFree Length . . . . . . . . . . . . . . . . . 6.5·10−3 m

ScopeScope . . . . . . . . . . . . . . . . . . . . . . . Body-Ground

MobileBehavior . . . . . . . . . . . . . . . . . . . . Deformable

Mesh

Body Sizing - Shim:

DefinitionType . . . . . . . . . . . . . . . . . . . . . . . . Element SizeElement Size . . . . . . . . . . . . . . . . 3.0·10−4 mBehavior . . . . . . . . . . . . . . . . . . . . Soft

Analysis Settings

Step ControlsAuto Time Stepping . . . . . . . . OnDefine By . . . . . . . . . . . . . . . . . . . SubstepsInitial Substeps . . . . . . . . . . . . . 1Minimum Substeps . . . . . . . . . . 1Maximum Substeps . . . . . . . . . 5Time Integration . . . . . . . . . . . . On

Solver ControlsSolver Type . . . . . . . . . . . . . . . . . IterativeLarge Deflections . . . . . . . . . . . . On

68 APPENDIX B. MODEL SETTINGS (360◦ CASE)

System Coupling Setup

Analysis Settings

Initialization ControlsCoupling Initialization . . . . . . Program Controlled

Duration ControlsDuration Defined By . . . . . . . . End TimeEnd Time . . . . . . . . . . . . . . . . . . . 0.011 s

Step ControlsStep Size . . . . . . . . . . . . . . . . . . . . 2.5·10−5 sMinimum Iterations . . . . . . . . . 2Maximum Iterations . . . . . . . . 5

Execution Control

Co-Sim. SequenceTransient Structural . . . . . . . . . 1Fluent . . . . . . . . . . . . . . . . . . . . . . 2

2-way fsi simulations on a shock absorber check valve839875/fulltext01.pdf · 2-way fsi simulations...

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