simulation of in compressible flow around rotating sphere

8/8/2019 Simulation of In Compressible Flow Around Rotating Sphere

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Simulation of IncompressibleFlow Around a Rotating Sphere

V I D A H E I D A R P O U R - D E H K O R D I

Master of Science ThesisStockholm, Sweden 2009


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Simulation of IncompressibleFlow Around a Rotating Sphere

V I D A H E I D A R P O U R - D E H K O R D I

Masters Thesis in Numerical Analysis (30 ECTS credits)

at the Scientific Computing International Master Program

Royal Institute of Technology year 2009

Supervisor at CSC was Johan HoffmanExaminer was Michael Hanke

TRITA-CSC-E 2009:097

ISRN-KTH/CSC/E--09/097--SE

ISSN-1653-5715

Royal Institute of Technology

School of Computer Science and Communication

KTH CSC

SE-100 44 Stockholm, Sweden

URL: www.csc.kth.se


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Abstract

Numerical computations of incompressible, inviscid flow at high Reynoldsnumber around a moving sphere, both spinning and non spinning, has beenpresented. Particularly we compute forces, i.e. drag and lift, on the ball aswell as modeling the wake and separation. The trajectory of the sphere hasbeen plotted and the effect of rotation and lift force on the trajectory has beenstudied. The G2 method with friction boundary conditions is used to computethe turbulent flow.We are testing both the method G2and the incompressible Navier-Stokes solveravailable for us. We show the computed results and how the solver has approx-imated the flow around a rotating sphere. G2 method that we use to computethe turbulent flow around a moving sphere gives a good prediction of the aero-

dynamic forces.The problem to compute turbulent flow in high Reynolds number is the very ex-pensive resolution of the turbulent boundary layer in high Reynolds caused bythe excessive number of mesh points required near the boundary. Even with to-days parallel computers, it is not possible to perform such computations. Thevariety of turbulent methods, e.g. LES, RANS, with different turbulent mod-els, e.g. wall model, Reynold stress, have been used to solve turbulent flows.The method we use to compute the flow around the sphere is G2 method withfriction boundary conditions. The G2method is a stabilized Galerkin finite ele-ment method using continues, piecewise linear trial functions in time and spacewith continues piecewise linear test functions in space and piecewise constanttest functions in time. The method uses to compute approximate solutions tothe Navier-Stokes equation. Instead of resolving the turbulent boundary layer,the method uses a friction boundary condition as a wall model.


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Sammanfattning

Simulering av inkompressibel strmning runt en roterande sfr

Numeriska berkningar av incompressibelt, icke-viskst flde vid hgaReynoldstal runt en boll presenteras, bade med och utan rotation. Specielltberknas krafter p bollen, med modellering avseparation och vak. Trajektori-an fr bollen har plottats och effekten av rotation med avseende p lyftkrafthar studerats. En G2-metod med friktionsrandvillkor anvnds fr att berknaturbulent flde.Vi testar bde metoden G2 och den fr oss tillgngliga lsaren fr de inkom-pressibla Navier-Stokes ekvationer.Metoden G2ger en bra frutsgelse av aero-dynamiska krafter. Detta r den metod vi anvnder fr att berkna turbulent

strmning runt en rrlig sfr.Problemet att berkna turbulent strmning fr hga Reynoldstal r att detr mycket dyrt att lsa upp det turbulenta grnsskiktet. ven med dagensparallella datorer r det inte mjligt att gra sdana berkningar. Diverse me-toder med turbulensmodellering har tidigare anvnts, t.ex. LES och RANS.I den metod vi anvnder hr lser vi inte upp turbulenta grnsskikt, istlletanvnder vi ett friktionsrandvillkor som vggmodell. Methoden G2 r ett sta-biliserat Galerkin finita element-metoden fortstter piecewise linjra rttegngfunktioner i tid och rum med fortstter piecewise linjra testa funktionerna irymden och piecewise konstant testa funktionerna i tid. Metoden anvnds fratt berkna ungefrliga lsningar till Navier-Stokes ekvation. Metoden anvn-der en friktion grnsen skick som en vgg modell, i stllet fr att lsa turbulent

boundary layer.


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Contents

Simulering av inkompressibel strmning runt en roterande sfr . . . iv

1 Introduction 1

2 Physical concepts of the model 5

2.1 Wake and separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Drag and drag crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Lift, Magnus effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Mathematics and Methodology 7

3.1 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 73.2 The boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Skin Friction Wall Model . . . . . . . . . . . . . . . . . . . . 8

3.2.2 Slip Boundary Condition . . . . . . . . . . . . . . . . . . . . 93.2.3 No Slip Boundary Condition . . . . . . . . . . . . . . . . . . 93.3 Other works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Some methods for turbulent flow . . . . . . . . . . . . . . . . . . . . 10

3.4.1 Discretization cG(1)cG(1) . . . . . . . . . . . . . . . . . . . . 123.5 Stability of G2 method . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Computation of forces . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.6.1 Computation of Drag . . . . . . . . . . . . . . . . . . . . . . 143.6.2 Computation of Lift . . . . . . . . . . . . . . . . . . . . . . . 153.6.3 Computation of Gravity . . . . . . . . . . . . . . . . . . . . . 15

3.7 Rotating ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.8 Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Computations 19

4.1 The Computational model . . . . . . . . . . . . . . . . . . . . . . . . 194.1.1 Still Ball Model . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 Rotating Ball Model . . . . . . . . . . . . . . . . . . . . . . . 204.1.3 Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 The Computational results . . . . . . . . . . . . . . . . . . . . . . . 224.2.1 Still Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.2 Rotating Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


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Chapter 1

Introduction

In this work we present numerical computations of the incompressible flow arounda moving sphere in air. Both cases of rotating sphere and non-rotating sphere havebeen considered. A prepossessing step will be taken by constructing a simple modelfor the geometry with the use of the COMSOL multiphysics software. The geometrywill be transformed to the solver. After the completion of the solving process of thepartial differential equation which is to be performed by the solver, the results havebeen presented and discussed using MATLAB, Paraview, and C++ programming.For this simple model, we are testing the specific solver used in our project and wewill investigate the outcomes as well as how qualitatively reasonable the results are.The model (the rotational moving sphere) which is a simple test case, is interesting

from different point of views. The particle transport in the engineering point ofview, or trajectories of spinning balls in sports. In every sports where a sports ballis being used, it is very important to realize the path in which the ball is travelingon.

Different methods, e.g. DNS (Direct Numerical Simulation), LES (Large EddySimulation), RANS (Reynolds Averaged Navier-Stokes), can be used to compute theturbulent flow. The mentioned methods shortly explained in Chapter 2. HoweverDNS is very expensive when it comes to solving the turbulent flow. The reason isthat DNS will fully resolve for all the physical scales of the model. In turbulentflow, the number of mesh points needed for full resolution are approximated to beRe3 [6] and a boundary layer around the moving object must be solved. Therefore

using DNS wont be a good idea. Most real life experiences are at higher Re. Wewill have turbulent flow in Re > 106, and what is obvious is that it is impossibleto use DNS with the computer systems available. Therefore, usually other methodsare used which all of them use turbulence modeling. Turbulence modeling is a widearea of research and different resolution methods have been offered to get aroundthe turbulent modeling problem. In this work we will just take a quick look atsome of the suggested methods and a little explanation of the method we used ( G2)and focus on the simulated results. Inspired by previous works done in [3, 6, 8]regarding bluff bodies moving through air to predict the mean value of drag and

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Introduction

lift for a non-rotating sphere, we will use the same method to model the rotating

sphere. In particular, an adaptive General Galerkin(G2) method with skin frictionboundary condition is used. We are using the icns, which is an incompressibleNavier-Stokes solver and is part of Unicorn solver that is developed by FEniCSprojects group, to simulate the turbulent flow around a rotating sphere. It isrelevant to mention that the G2 method was implemented to the incompressibleNavier-Stokes solver which we are using. This means choosing this solver, we areusing the G2 method automatically to compute the approximate values of interest.There is a brief explanation in the following chapters of this project about the G2method and skin friction boundary conditions. However, more specific informationabout the method can be found in [3, 4, 6, 8].

G2 doesnt resolve the boundary layer to compute the mean value of drag and

lift of turbulent flow. We get a good approximation of wake, separation point(s),and mean value forces. In [8], we notice that an adaptive G2 method is based ona posteriori error estimate which is the combination of the residual, and the dualweight.

The FEniCS Project is a set of free software projects with the common goal toenable automated solution of differential equations. It has many components andone of them is the Unicorn solver which is an adaptive finite element solver forfluid and structure mechanics that uses G2. Unicorn is based on the python/C++interface of FEniCS called Dolfin. Dolfinprovides an environment for solving partialand ordinary differential equations1.

The investigation of flow around a sphere can be an interesting subject in many

fields of studies such as in CFD, sports,etc. It is important in many sports to givethe sports ball a spin that it needs to change the straight path that it is movingin, e.g. both in topspin and backspin in table tennis.

Also, this is an interesting test case to test the performance of the solver in use.The computations we are doing are also important scientifically speaking becausethe results of these computation can support the new resolution of dAlambertsparadox, which was given by Claes Johnson and Johan Hoffman [9].

In this work we shall show that one can predict mean values of interest forrotating sphere in small viscosity flows without resolving boundary layers, usingAdaptive G2 with slip and friction boundary conditions.

The flow separation for circular shapes depends on the Reynolds number. There-

fore, in modeling the sphere it is interesting to see how the separation point(s) andthe wake and therefore the drag will act due to changes of Re. We also can checkthe accuracy of the method comparing the results from the wind tunnel experi-ments with the results from the solver or we can study the computational cost ofthe algorithm we are using, e.g. computing time and memory requirement.

In the model, skin friction (noted as parameter) is coupled to energy dissi-pation. Increasing the skin friction at the boundary will cause the momentum todecrease which will result in an earlier separation and a bigger wake with an increase

1For more information about the solver refer to http://www.fenics.org

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Introduction

in drag. This means that friction parameter is coupled to velocity (velocity Re).

An increase in will model a decrease in Re as well as the velocity of the spherewhile it moves in the air. The separation point(s) of circular shapes objects suchas sphere are related closely to Re so that in our experiments by changing , wenote a sudden change of the separation points (and wake and drag). We connectthis behavior to the drag crisis of Re 105 which corresponds to modeling theboundary layer as it is changing from turbulent to laminar flow. This can be con-cluded from the previous experimental results in [6]. A brief conclusion from thementioned work is that for a laminar boundary layer, i.e. 103 Re 105, thepositions in which the flow will separate from the object stay almost the same withno big change. However, as the boundary layer changes to turbulent, i.e. 105 Re,a visible delay on the separation of the flow from the boundary will occur.

In Chapter 2, the physical explanation of the model is presented. Concepts such asDrag crisis, Magnus effect and boundary layer theory have been presented in thischapter. In Chapter 3, we give a summary of the history of turbulent flow and com-putational methods as well as an explanation about the mathematical formulationwe used to solve the problem. Also, a summary about the method used and thewell-posedness and stability of the method is given. In Chapter 4, The computa-tional model and the computational results as well as the conclusion is presented.First, We consider the problem of a non-rotating sphere with several different fric-tion coefficients, in different simulations. Then we will compute the flow around arotating ball (clock-wise and counter clock-wise) with friction boundary conditionwhere the skin friction is modeled to change linearly during time. Computational

models and results will be presented. At the end of this chapter the Conclusion andfuture work will be presented.

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Chapter 2

Physical concepts of the model

In this chapter, I would like to explain some of the physical concepts that we willuse in this paper.

2.1 Wake and separation

In CFD, the separation point(s) are the points in which the fluid that is moving tan-gentially to the boundary separate from the boundary. The flow tends to separateearlier for a laminar boundary layer with low tangential velocity near the boundary.The track of turbulence that is left behind the moving object in the air is called

wake.

2.2 Drag and drag crisis

Drag is an aerodynamic force. Aerodynamic forces arise from two causes: (i) pres-sure on the surface of the object, (ii) viscosity (skin friction). Drag force will begenerated when a sphere shaped object such as a sport ball is moving in air due to(i), (ii). This force will be in the opposite direction to the moving ball and will resistwith the movement of the ball which will cause the velocity of the ball to drop. Forhigh Reynolds numbers, the skin friction plays a small part in determining the dragforce and instead the most determining drag force is the one due to pressure which

is coupling to separation of the flow. In high Re, one can explain the drag force bythe separation point(s). According to [7] the earlier the separation, the higher thepressure drag, and therefore is the drag force.

While the delay in separation will cause a smaller wake behind the ball andthe drag force will drop noticeably and the ball will be permitted to travel furtherthan regularly expected. This sudden, high reduction of drag force is known as dragcrisis.

Drag crisis happens for 105 Re 106. This interval happens to correspondto when a laminar boundary layer undergoes transition to become turbulent. Theincrease of momentum of the turbulent boundary layer will cause the delay in sep-

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Physical concepts of the model

aration and therefore less drag force will be generated. One physical example of

drag crisis is the dimples of a golf ball. The effect of these dimples is to produceturbulent boundary layers so that the ball can travel further with a moderate veloc-ity (no need of high Re to produce turbulent boundary layer), according to Prandtland Wieselsbergers experiments.

2.3 Lift, Magnus effect

When a sport ball rotates around its axis, in addition to its straight forward motion(which will produce the drag ), another aerodynamic force called the Lift force isalso produced which is prependicular to the direction of the rotation which dependson the angular velocity of the sphere as well as the direction of the movement. Thelift force is force that is perpendicular to the flow direction. It contrasts with thedrag force, which is parallel to the flow direction. The effect of the lift force onthe rotating object is known as Magnus effect or reverse Magnus effect dependingon the direction of rotation.The expression of the Magnus effect comes from LordRayleigh, who credited Heinrich Gustav Magnus for the first explanation of thelateral deflection of a spinning ball [7].In most sports played by ball, these effects will play a major role in changing thescore of the game by changing the path of the ball1. The physical example andexplanation of Magnus effect and reverse Magnus effect:1) The traditional explanation is based on Bernoullis law stating that the reasonof Magnus effect is the pressure difference between two sides of the ball caused byrelative velocity vectors on each side. It claims that on the side of the ball whichis moving in the same direction of wind flow, an increase of the velocity (decreaseof pressure) will be generated and on the other side which has a velocity vectoropposite to the wind flow, a decrease in that side will be observed. Moreover, thepressure difference will cause the ball to move toward the side that has a lowerpressure.

2) A more modern understanding of the Magnus effect is attributed to asymmet-ric separation in turbulent boundary layers due to different relative velocities, seee.g. [7]. In this approach the separation points are used to explain the phenomenonand basically the cause mentioned in this approach is the horizontally unsymmet-rical separation. This is when the delay in separation will cause turbulence in thatside as well as producing the other side to be laminar due to premature separation.As the turbulent flow is relative to the increase of the momentum, this additionalmomentum should transfer to work (here it is the generation of Magnus force).Using this notion, it is possible to explain the topspin and backspin in tennis game.

1 An example is the free-kick of Roberto Carlos against France in 1997.

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Chapter 3

Mathematics and Methodology

One can use the familiar Navier Stokes (N.S.) equations which were formulated1825 45 to compute the fluid flow. These equations are widely used inComputational Fluid Dynamics (CFD) for computing both laminar and turbulentflow. We are going to use them for our computations of the sports ball as well.The N.S. equations are nonlinear partial differential equations in almost every realsituation, with the exceptions of one dimensional flow, and Stokes (creeping) flow.The nonlinearity makes most problems difficult or impossible to solve.

3.1 The Navier-Stokes equations

The most general Navier-Stokes equation end up being like (3.1), which is indicatingthe conservation of momentum. The second equation is the Continuity equationandhere it indicates the conservation of mass.

u

t+ u u

= p + T + f (3.1)

t+ (u) = 0. (3.2)

Here u is the flow velocity vector, is the fluid density, p is the pressure, T is thedeviatoric stress tensor, and describes viscous forces, and the stress tensor is definedby ij = pI + T.T in equation(3.1) has too many unknowns which makes the original N.S. not thatusefull for solving problems. Instead assumptions will be made on viscous behaviorwhich will eventually give us a useful equation. For incompressible, Newtonian flow T = u.For incompressible, Newtonian, with unit density ( = 1) fluid in volume the

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The boundary condition model that has been used is as follow:

u n + nTn = 0, (3.4)

u k + 1nTk = 0, k = 1, 2 (3.5)

These equations are implemented weakly to the discrete equation of cG(1)cG(1) [7]and this boundary condition was implemented to the mesh points on the boundaryof the sphere. More details on this boundary condition can be found in [5]. Thefirst equation is used to define the penetration of the boundary by modifying ,which in our case is always zero since the sphere is solid and the flow will not gothrough it. The second equation implements friction by the parameter whichcorresponds to slip b.c. with = 0 and nonslip b.c. with . is basedon: Re, the roughness of the body of the sphere (skin friction ), etc. Increasing will cause an increase in the friction of the boundary, while decreasing will modelreduced (normalized) friction which will cause an increase of velocity and thereforeincreasing Re.

Velocity and length are both scaled to be of unit size in the computations andusing we are able to model the velocity. Choosing to be a function of velocity intime will couple the value of beta to velocity and therefore to the Reynolds number.By increasing we model the velocity (Re number) of the sphere being decreased asit moves in the air. Re is coupled to velocity with Re = U L , where in the physical

scale L is the actual diameter of sphere and it is fixed in time. is the kinematicviscosity of air ( 105m2/s) which is constant in time, assuming the temperatureto be constant (kinematic viscosity depends mostly on temperature).

3.2.2 Slip Boundary Condition

A slip boundary condition is when there is no friction or resistance of the boundaryof the object against the flow moving on it, another way of saying is when the profileof velocity will not change along the boundary. It can been shown with the followingmathematical term : u n = 0. Using = 0 in formula (2.5) will supply the slipboundary. By using Adaptive G2 method with slip boundary condition, one is ableto solve the Euler equation. In the solver for incompressible Navier Stokes, availablein Unicorn we declare the viscosity term in Navier Stokes to be zero. Since in thecase that boundary condition is slip (and the skin friction will vanish too), therewill be no artificial viscosity. In this case we will be solving the Euler equations.

3.2.3 No Slip Boundary Condition

No slip conditions refer to the state in which the flow will not slide on the boundaryof the object but instead the flow particles will stick to the boundary. This is whenthe profile of velocity reduces as it gets closer to the boundary, and is zero at theboundary. This simply means u = 0 at the boundary of the object.

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In our method, varying the skin friction parameter () means that we are using

a skin friction wall model boundary condition around the sphere. To implementno slip boundary condition as the boundary layer approximation condition of themodel, we can simply give = 1 in the solver.

3.3 Other works

There has been a lot of work done previously for a non-spinning sphere in whichthe drag has been computed. In [6], G2 method has been used for non-rotatingball. In [12], another method for turbulent modeling called DES has been used tocompute drag crisis. Detached Eddy Simulation (DES) is a method that acts as

RANS near the boundary and changes to LES a way from the surface.Incompressible, viscous flow past a rotating sphere (with rotation around thestream line axis), for low Re (50 Re 100), has been done in [2] in which a three-dimensional spectral element/spectral method (combination of a FEM method withhigh degree piecewise polynomial basis and spectral method known also as FourierTransform Methods) has been used. The same method was used in [1] to computethe lift and drag of the rotating sphere in laminar flow.

In [2], the point potential vortex method has been suggested for a rotating ballmoving in an incompressible, inviscid flow as well as the numerical computation ofthe vortex sheet in the a plane.

In [11], flow around a rotating baseball was computed using a combination of

types of finite difference methods in space and time. From their conclusions: (i)it appears that the variations in the wake become larger when the speed in whichthe ball spins is lower. (ii) The drag is predicted to increase as the rotating speeddecreases which explains the Magnus effect and why the topspin will land sooneron the ground.

In most methods the boundary layer theory has been used in which a viscous flowin a boundary layer must be solved which needs modeling and is very complicated.

3.4 Some methods for turbulent flow

It is important to predict a reasonably accurate solution to the N.S. for turbulent

flow since most fluid flows in real life are turbulence, e.g. blood flow in arteries, oiltransport in pipelines, atmosphere and ocean currents, and the flow in boat wakes,around aircraft-wing tips and along the sport ball moving in high velocities, etc.

The variety of length scales needed to be resolved in order to give an accurateestimation of the turbulent flow will cause to fail computing turbulent flows point-wisely. Another way to explain it is to say that a very fine mesh is needed in orderto capture all these length scales.

One of the basic problems in CFD area is to compute the turbulent fluid flowaround a moving object. According to [6], the number of mesh points needed forsolving turbulent flow has been approximated to be Re3 in space-time, where Re =

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UL/ and U, L are scaled velocity and scaled length of the object respectively

and is the viscosity of the fluid. Computing the turbulent flow with large number ofdegrees of freedom close to the boundary layer is a challenge. Several methods havebeen suggested for turbulent modeling. One of them is DNS which is a numericalmethod used to solve N.S. equations without modeling. It will resolve for all physicalscales of flow and the cost will grow fast with increasing Re and it can not be usedfor high Re flows. For most cases of interest (i.e. Re > 106), the number of degreesof freedom in space-time that are needed to resolve all scales of the flow will be1018. This is not computable even with todays massive parallel computers.

Using the laminar solver to model the turbulent flow has failed as it becomesunstable (not converging) in time. Therefore the idea of using a time-averaged N.S.instead of the real N.S. equations gave raise to some methods like RANS, LES,...

Thereby different methods has been proposed to get around the problem inwhich most of them use turbulent modeling which will introduce a model usuallyusing an averaged N.S. equation. Two most common method used to get aroundthis problem are RANS, Reynolds Averaged Navier Stokes, and LES, Large EddySimulations. In the first method, the effort is to solve a modified (averaged) NavierStokes equations, with some approximate solution which will usually end up addinga term called Reynolds stress, to the modified equation. This comes down to RSM(Reynolds Stress Modeling) which requires a complex modeling. The idea in thismethod is based on Reynolds averaging in 1894.

Formulated in late 1960s, LES uses the idea to only solve for large eddies ofturbulent flow and using a subgrid model to model the smaller eddies. The idea

behind this method comes from the self similarity theory formulated on 1941 byAndrey Kolmogorov.

In self similarity theory one assumes that large eddies of the flow are dependenton the flow geometry, while smaller eddies are self similar and have a universalcharacter. Many methods have been produced by combining the above methodstogether at which all will search for approximate solution to an averaged (filtered)N.S. equation and will use turbulence modeling which are complicated. Instead inG2 we will compute weak solutions (more specifically, CU||hR(U)||-weak solution,explained in [7]) to N.S. equations. In this method we dont resolve all physicalscales as oppose to DNS. Also, we wont resolve turbulent boundary layer so thereis no need for turbulence modeling based on physics of unresolved scales as oppose

to other turbulent modelings . Looking for -weak solution instead of an averaged(filtered) N.S formula we have no turbulent model (such as subgrid modeling orReynolds stress modeling) but more to say using G2 we will get an automaticturbulent model. That is to say we will model the turbulent boundary layer usinga slip with friction boundary condition which is discussed in boundary conditionsection. To solve the laminar boundary layer (which happens for Re < 105) themethod uses N.S. equation with either noslip or high friction boundary conditions.

The -weak solution is an expression used in [7] for the solution of the methodin use. That is to say the residual control will be ||R(u)||1 < instead of the morecommon residual control ||R(u)|| < . The weak norm ||.||1 is defined by:

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||w||1 = supvH1

0

v.wdx

||v||H1(),

||w||H1() = (||w||2 + ||w||2)1/2

According to [7] the residual will be very small for turbulent solutions in the weaksense while it is very large pointwisely. This is the reason that the method canbe used to compute some mean values of interest. We display the mean value ofinterest with a function called M(u).

To discretize the system of equations, we are using the weighted least square

stabilized cG(1)cG(1) method. The stabilized cG(1)cG(1) method is a type of G2method that uses continuous stabilized Galerkin finite element method in time withpiecewise linear trial functions both in time and space and linear piecewise testfunctions in space and constant piecewise test functions in time.

The stabilization term in the discretized system will introduce dissipation wherethe residual is large. The adaptivity is based on a posteriori error estimation of thedual problem with respect to a function of drag. According to [7], using somefunction of the solution (i.e. drag, lift, etc.) will give rather more satisfying boundsof the error whereas the energy norm in most cases doesnt give good bounds,compared to what is acceptable in reality of physics.

3.4.1 Discretization cG(1)cG(1)The method has been explained in [8]. Here is just the summary review of thesection.

We are looking for approximate solutions for velocity (Unh Uh(tn)) and pres-sure(Pnh Ph(tn)) which will satisfy N.S. equations with homogeneous Dirichlet bound-ary conditions. Let 0 = t0 < t1 < ... < tN = T be a sequence of discrete time stepswhere the time interval at each step is In = (tn1, tn] with length kn = tn tn1.Assuming the domain to be Sn = In and let Wn H1() be a finite ele-ment space of continuous piecewise linear functions on a mesh of size hn with thefunctions in Wn satisfying the Dirichlet boundary. H here is a Sobolev Hilbert

space. Uh(tn) V0n [W0n]3 and Ph(tn) Wn. functions in the set V0n satis-fying the Dirichlet boundary condition v|D = 0 and functions belong to set Wn

satisfy the Dirichlet boundary condition v|D = w. Choosing the test functionsv V0

n H1(), the numerical approximation of the discretized system is definedby:

((Unh Un1h )k

1n , v) + (U

nh U

nh , v) + ( 2(U

nh ), (v)) (P

nh , v) + ( U

nh , q)

+ SD(Un, Pn; v, q) = (f, v + 1(U

nh v + q)), (v; q) V

n0 W

n,

where

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SD(Un, Pn; v, q) = +(1(U

nh U

nh + P

nh ), U

nh v + q) + 2 U

nh , v)

with Unh =12(U

nh + U

n1h ), 1 =

12(kn

2 + |U|2h2n ) 1

2 and 2 = k2h. k1, k2 arepositive constants of unit size.

(v, w) =

Kn

K

v wdx,

((v), (w)) =3

i,j=1

(ij(v), ij(w)).

3.5 Stability of G2 methodthe stability estimation has been proven in [7]. The following relations holds forf = 0;

1

2 U(t) 2 + h1/2R(U) 20=

1

2 U0 2

Where R(u) = u + U u + q and u = (u, q). This shows that the numericaldissipation in G2 is proportional to the residual R(U). Also, it determines thatresidual will not grow and therefore we have stability in the form,

h1/2R(U) 02

12

U0 2

3.6 Computation of forces

According to [3], the mean value forces effecting an object situated in air over timeinterval (I) could be computed by following formula,

N((u)) =1

|I|

I

(u + u u f, ) (p, ) + (2(u), ()) + ( u, )dt

where u = (u, p) is the -weak solution to the N.S. equations. is a function defined in the fluid volume to be a unit vector in the direction

of the force that we want to compute in 0 and to be zero on the rest of the boundary1 = /0. 0 is the part of the boundary that is in contact with the flow andthe remain will be 1. In this model (a sphere moving in air) all the boundary willbe in contact with the flow. Therefore, it is important to know the direction ofdifferent forces acting on the sphere in order to compute them.

This formula depends on , and the distance between and the boundary.Using a finite element method cG(1)cG(1), we are able to approximate the force inthe discretized system as follow,

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Nh((Uh)) =1|I| I(Uh + Uh Uh f, ) (Ph, ) + (2(Uh), ())+

( Uh, ) + SD(Uh, Ph; , )dt.Now and are finite element functions. SD is the stabilization term. Uh =Uh

nUhn1

knon In (In, is the nth subinterval of the total time interval I).

There are three aerodynamic forces on the ball moving in the air. In following thecomputations of the forces on the ball (i.e, drag, lift, gravity,etc.) will be presented.

3.6.1 Computation of Drag

The drag force is the force that resists the movement of a solid object through air.It is parallel and opposite to the direction in which the object is moving (moreinformation on drag can be fOUnd in next chapter).

To compute drag force it is only needed to define = (0, 0, 1), assuming the ballis moving in x3 direction. According to the computations in [7], with considerationof a longer time interval we are able to compute the mean drag force more accurate.The drag coefficient (cD) is a dimensionless quantity which is used to quantify thedrag or resistance of an object in a fluid such as air. We are interested to computethis coefficient. Under the assumption that Re is the same and the flow is subsonicand steady, cD is the same even if other variables change. As a test case we havethe drag coefficient computed for sphere to be approximately 0.4 for a rough spherein Re = 106. Even though Re and therefore cD can vary a lot, it is almost constantor very small in some range of interest.

Figure 3.1., by Auchenbach (1974) [10], demonstrates the variation of cD

as afunction of the Reynolds number (for different smoothness of sports ball).

Figure 3.1. Aerodynamic drag curves for spheres of different roughness, k/d = 0,25, 150, 250, 500, 1250 E-5 increasing to the left

We will then compute the mean value of the drag coefficients during the time

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interval I = [0, T] (e.g, cD =1|I| I cD) for incompressible flow by the following for-mula [8, 7, 3],

cD N((u))12U

2A(3.6)

where the U = 1 is the normalized velocity, and A is the cross-sectional area ofthe object which in our study case for a sphere is ( D2 )

2, with D the diameter ofthe sphere. N((u)) is the mean value drag force over time interval I.

3.6.2 Computation of Lift

The lift force is another force which might be acting on the moving object on theair. that is perpendicular to the flow direction. It contrasts with the drag force,which is the parallel to the flow direction. We will compute the mean value of thelift coefficients during time interval I = [0, T] (e.g, cL =

1|I|

I cL) for incompressible

flow by the following formula [8, 7, 3],

cL N((u))12U

2A

In this case we will define N((u)) to be lift force by defining = (0, 1, 0) if the liftforce acts in x2 direction.

3.6.3 Computation of Gravity

Another force that we can consider is earths gravity which will be perpendicular tothe ground and is always downward assuming the origin of the coordinate systemto be the origin of the earth (g 9.8 m/s2).

However, in our solver the coordinate system has been transformed with theaxes defined by the direction of the movement of the sphere; Therefore, the gravityforce direction will depend on the place of the sphere in the global coordinate system

at each time. To model the gravity force, we will need to know the direction of thegravity force at each time step which is not always downward anymore but changesfrom {i}

i=0n1. n = (tn) is the angle between gravity and drag force (or opposite

direction velocity flow) at time step n. Note that 0 n .Therefore having n, we can easily compute the gravity force at each time step

to be,

Gh((tn)) = g

0cos (tn)sin (tn)

.

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It is enough though to define = (0,cos(n),sin(n)) in the solver, if the displace-

ment of the ball is in x2, x3 directions and there is no displacement in the directionof x1(the sphere is only rotating around the x1 axis). Note that we will add thegravity to plot the trajectory of the sphere, and since the first component is zerowe can say that the computations are in 2D. However all the other computationswhich are done in the solver are in 3D.

Now the question is how to compute the n at each time steps. Under the as-sumption that the sphere is moving along the negative direction of x3 axis and thepositive direction of x2 axis they can be computed by,

n = arctan(Xn+1(3) Xn(3)

Xn+1(2) Xn(2))tan n 0

n =

2+ arctan(

Xn(2) Xn+1(2)

Xn(3) Xn+1(3))tan n > 0

where Xn is the global position of the ball at time n. This will be computed by thefollowing explicit system of equations.mX = F,which can be taken as two first order following equations:mX = U,U = F.Here, m is the mass of the ball, F is the resultant of the forces on the ball. This canbe added to the solver to compute gravity force and adding it to the aerodynamic

forces in order to plot the trajectory online. However, we did not use this for ourcomputations. Both the gravity force and plotting the trajectory are done as anoffline operation.

3.7 Rotating ball

For an object to be able to rotate, a force is needed. The force needed for rotationwill be produced by angular velocity. We can compute the angular velocity in whichthe ball is rotating clockwise around the x1 axis as follow:

uball = x , (3.7)

Where is the angular speed (rotational speed), x is the distance vector fromthe rotational axis (x1) to the surface of the ball which will differ with

x . Assumeprojecting the sphere on the x2-x3 plane, then we will have different isosurfaces(circles) with different radius(

(x2)2 + (x3)2). We will compute the above formula

for counter clockwise rotation (corresponding to topspin),

=

00

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x = x1x2

x3

uball =

0x3

x2

The sign of will change for backspin. In our computations to get |uball|max =1/2|uair| we choose

= uair/2R where R =

(x1)2 + (x2)2 + (x3)2 = rmax. rmaxis the radius of the biggest isosurface.

3.8 Trajectory

One interesting concept is to see how the rotation of the ball will effect the curvatureof its movement. Therefore, we compare the trajectory of the ball in three cases of:(i) topspin (ii) nospin (iii) backspin. The position of the ball and where it is goingat each time step will be needed in order to plot the trajectory of the sphere. X,the position of the ball is computable from a simple second order partial diferentialequation (this is known as Newtons second laws of motion, or laws of acceleration).F = mX where F is the resultant of the forces on the ball and m is the mass of theball (depends on the type of the ball we use). F contains gravity force as well asthe aerodynamic forces, i.e. drag and lift. Additionally, the initial position of thesphere X0 and initial velocity of the sphere U0 will be given. The trajectory of theball can be obtained by the following discrete system,

Xn = Xn1 + k Un1

Un = Un1 + (k/m) Fn1, n = 1,...,N

To track the position of the ball which is needed to plot the trajectory, we couldeither add it within the solver or we could do it offline. In our work, we chose theoffline routine because it was easy to implement and since we have had run ourcomputations and already had the forces on the ball therefore we didnt want to

run the whole computations all over again, it seemed more efficient for us to chosethis option.

Computations ofF can be done using a rotation matrix to interpolate the aero-dynamic forces to a chosen global coordinate system where the gravity force will bealways in the opposite direction of the x2. In this way, at each time step we havethe aerodynamic forces from the solver and we get their direction using the rotationmatrix. Another way is that we can use the rotation matrix to interpolate gravityforce at each time step and we can add that easily to the solver. This way the direc-tion of the aerodynamic forces are fixed and the gravity force will be interpolatedaccordingly to the coordinate system used by solver, at each time step. I didnt

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use the rotation matrix. Assume that in the global coordinate system the gravity

force to be downwards at all time, the drag force to be parallel and in the oppositedirection to the velocity at each time step, and the lift force to be perpendicularto velocity at each time step. Having the initial velocity v0 and using the velocitywhich is estimated at each time step by the above formula which is dependent on theforces at each time step, we can find out the position of the ball in our coordinatesystem and plot the trajectory.

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Chapter 4

Computations

The model we are studying is a sphere situated in a box. We used the ComsolMultiphysics environment in order to define the geometry of the model as well asgenerating the initial mesh.Comsol Multiphysics is an environment which may beused to define the geometry, specify the physics of the model, mesh generation, solveand post-processing the results. After generating the geometry and the initial meshusing Comsol, we will solve the system using icns (Incompressible Navier-Stokessolver) to solve the discrete system, and an adaptive mesh refinement algorithm torefine the mesh which are both available in Unicorn package. The postprocessinghas been done in Paraview and MATLAB. ParaView is an open source package usedfor data analysis and visualization1. We used it to visualize the solution and used

MATLAB in order to plot the related graphs. MATLAB is a high-level language andinteractive environment that enables you to perform computationally intensive tasksfaster than with traditional programming languages such as C, C++, and Fortran2.It can be used for computational, visualizations, and programming purposes. Itis easy to use since problems and solutions are expressed in familiar mathematicalnotation

4.1 The Computational model

4.1.1 Still Ball Model

The G2 method was used in [6] where the computations of mean drag value in timeas well as visualization of separation point(s) for a moving ball in air (changing thevelocity of the ball by taking the friction parameter to be a function of velocity)have been presented. The results showed that when decreasing the friction param-eter (increasing the velocity) from = 0.1 (correspond to the laminar boundarylayers) the drag almost remains constant until in = 0.02 when laminar boundarychanges to turbulent a sudden drop in drag happens (drag crisis). In this work, we

1http://www.paraview.org2http://www.mathworks.com/products/matlab

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take the velocity variable fixed so that we can see how the smoothness of the ball

will effect on drag, and separation point(s) (skin friction is a function of velocityand smoothness, etc.). An experiment has been done in [10] regarding the differ-ent sports ball with different smoothness which gave different drag coefficient. Intheir work they have used a radar speed gun downloading at 33 Hz to a notebookcomputer in order to track the fall velocities. There are also experiments done inwind tunnel for different ball skins. The result we show in this work are capturedusing the G2 method. For modeling the flow around any sport ball through air, weuse G2 with friction boundary condition which is available in icns (IncompressibleNavier-Stokes Solver) implemented in Unicorn software. According to [7], Prandtland Wieselsberger showed in a classical experiment that the properties of sportballs can effect the volume of wake behind the ball. As a result in same condi-

tions, different sport balls will produce different drags. In particular, consideringthe smoothness which is coupled to skin friction of the ball to differ from smooth tofuzzy skins and computing the drag force and separation of different ball surfaces.As mentioned before, drag crisis will happen when the laminar boundary layer willchange to become turbulent. This will happen for 105 Re 106. In high Re wewill have turbulent boundary layer.

Taking the skin friction () to be a function of velocity, one can model a movingball while the velocity differs. We can control the velocity by . Since the velocityU and the Reynolds number Re are related with Re = U L (knowing the geometryof the sphere will not change and assuming the kinematic viscosity to stay constantas the ball is moving.), it is easy to model a turbulent or laminar flow by choosing

the depending on velocity.In the result section of this chapter screen shots of separation points and wake as

well as solutions of velocity and pressure will be presented. The separation pointswill be presented using glyphs. There are plots showing how the drag force willchange relative to time and relative to mesh points (or level of refinement).

The model:

A sphere with D = 1 centered at x = (5, 5, 5) in the channel of dimension10 15 15 oriented in x3 direction, subject to inflow velocity (0, 0, 1) has beenmodeled. We are using a slip boundary condition on channel walls, Dirichlet inflowboundary condition, and transparent outflow.

4.1.2 Rotating Ball Model

To model a spinning ball we consider two different models: (i) modeling topspin,(ii) modeling backspin. It is easy to model both topspin and backspin, using icns.A simple modification of the wall function which was used previously in [6] to modelthe skin friction boundary, will suffice. The angular velocity of the ball uball, whichwill cause the spinning, can be implemented using the following wall model. Boththe angular velocity and wind velocity are controlled by friction parameter in the

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following formula,

(u uball) n + nTn = 0, (4.1)

(u uball) k + 1nTk = 0, k = 1, 2. (4.2)

Using the same geometry and mesh as used in still ball we add the angular velocityby above formula. We modify our code to model for the varying velocity andangular velocity in a simulations. We use scaled velocity which can be controlledby the friction parameter . In here taking to increase in time corresponds todecreasing the velocity. For topspin, the ball is rotating around x1 axis clockwisewith a dimensionless angular speed (nondimensional rotation rate, or the ratio of

the surface speed and free-stream speed) = |(uball)max||uair| = 1/2.

For the backspin, we use a counter clockwise direction for the rotation, everything else is the same.

4.1.3 Trajectory

The spheres trajectories are computed for different experimental problems. As-suming the initial position X0 = (0, 0, 0), two different cases of sphere diameters (d)have been experimented with different initial velocity in which the sphere has beenthrown. The experimental cases are as follows and the results for trajectories willbe presented at the results section;1) d = 1 : u0 = (0, 1, 1), u0 = (0, 50, 1), u0 = (0, 300, 1), u0 = (0, 20, 1)

2) d = 0.1 :u0 = (0, 1, 1), u0 = (0, 30, 5), u0 = (0, 10, 5), u0 = (0, 50, 20),u0 = (0, 4, 6)

The non-dimensional angular velocity is 0.5 for both topspin and back spin. Inorder to draw the trajectory of the ball, one can use one of the following alternatives:

(i) Online: It is easy to improve the model of the ball by changing dynamicallybased on the actual velocity of the ball. (extending the model from the assumptionof linear increase in ). It can be done by defining and adding the gravity force inthe solver. This can be done the same way as the drag and lift forces definitions inthe solver. It will be done by giving the force direction. The only difference here issince the mesh is not moving or say the local coordinate system is fixed on the mesh.Drag is always in the direction of the flow and lift is perpendicular to the flow, in

the local system. Using this method, the direction of the forces will not change intime stepping. However, the direction of the gravity force will change relative tothe flow direction during time steps. The angle between gravity and flow will be afunction of time which will vary almost by on angle of . (|(ti)| ). Varyingfrom acute to obtuse angle or vice versa depending on the direction in which theball is traveling. There is more information on how to compute the direction ofgravity force of the model in section 3.7.3.

(ii) Offline: The second alternative is to do the computations necessary to plotthe trajectory outside the solver. Using Newtons second formula mX = F. Writinga simple C++ program to solve the explicit system of equations, we can find the

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position (X) of the sphere at each time and we can easily plot the trajectory using

MATLAB. The mentioned system of equation is a second order ODE (ordinarydifferential equation) which can be solved by breaking it into two first order ODE.At each time step we take the approximate forces on the sphere (drag and lift)which are computed by the icns solver.In order to plot the trajectory we used the second method.

4.2 The Computational results

Please note that except the trajectories, all the results are computed in three di-mension.

4.2.1 Still Ball

With different skin friction and different mesh refinements we experiment that atthe beginning of the time interval, behavior of Fd is irregular but after a while thechanges of Fd during time becomes more regular and the range of oscillations willbe within an specific interval; Therefore, the mean value of Fd during time will beconstant, computable and meaningful. The following plot is an example of how thedrag force will look like at the beginning of time interval (left) and at the end of timeinterval (right), for a test case with skin friction = 0.5 in 5th mesh refinementwith 15872 number of mesh points.

Figure 4.1. = 0.5 : Drag force (Fd) vs. time

We know that the expected drag coefficient for a non rotating sphere moving inthe air is approximately Cd = 0.4. Considering this and the fact that Cd 2Fd willgive the conclusion that Fd 0.2. The following plot shows that as we refine themesh, we get closer to the expected drag force and therefore for accurate refinedmeshes we expect to quantitatively predict the value of drag force, using the solver.All the above stands for different skin frictions test cases. However, the followingshows the case where = 0.5.

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Figure 4.2. = 0.5:Drag force vs. time for in different mesh refinements.

In the simulations for slip boundary condition (corresponding to = 0), theamount of Cd is close to the mentioned expected value. However for a sphere withskin friction, more simulation (more refinement of the mesh) is needed to reachto the more accurate value. The plot in the right (corresponding to = 0.01)

shows that the function is decreasing in time. Quantitatively speaking, the valueof functions will converge to the predicted value with further mesh refinements.The following plots show the drag coefficient for still ball in two test cases whichdemonstrates Cd is reasonably close to previous studies.

Figure 4.3. cd vs. loglog number of mesh points for = 0, = 0.01 respectivelythe left and the right.

We can get a more accurate and smoother function by adding less points at eachrefinement of the mesh, if that is an interest of us. However, it was not a concern

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Computations

for us in this work. The following plot shows how different refinements will effect

the plots. For this test we studied the sphere with skin friction of0.01 for two cases.One with 10% mesh refinement the other with 5% mesh refinement. The followingplot demonstrates the zoomed in statue of mean value cd vs. loglog number of meshpoints (right). The right plot shows from 6th to 12th refinement and in the left plotwe have considered the initial mesh up to the 12th refinement.

Figure 4.4. = 0.01:cd vs. loglog number of mesh points

In the following plot the mean value of drag coefficient against different values ofskin frictions has exhibited. We considered this experiment for the mesh with 15872nodes in the 5th iteration of mesh refinement. It shows that for 0 0.05 wehave a dramatic change of drag coefficient in compare with the larger skin frictionand as the boundary converges to be no slip.

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Figure 4.5. cd vs. at 5th refinement

screen shots of solutions with three different skin frictions has been shown inthe following plots. For = 0 (zero friction at the boundary) the delayed in theseparation can be seen. Also, for = 0.01, = 0.5 the separation points and wakeare presented.

Figure 4.6. = 0 : Wake (left), separation points (right) in mesh with 20116 nodes

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Computations

Figure 4.7. = 0.01 : Wake (left), separation points (right) in mesh with 66082nodes

Figure 4.8. = 0.5 : Wake (left), separation points (right) in mesh with 66082nodes

Here is just three screen shoots of how the solution looks like in different timesteps in one simulation.

Figure 4.9. = 0.5 : Velocity in three samples in time

In the following the solutions of last time sample has been shown. The firstfigure belongs to = 0.5 with 15872 mesh points and the second one belongs to = 0.01 with 66082 mesh points. In the solutions it is apparent that the velocityand pressure of the flow are horizontally symmetric. There is no lift force. pressureis low at the separation points while velocity is high. The flow in second figure ismore mature in a sense that the mesh is finer. In the third figure that belongs to

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= 0 and 20116 mesh points, the substantial drag is noticeable. The pressure and

velocity are not vertically symmetric and therefore the drag wont be zero. Insteadwe are dealing with substantial drag which will be more obvious in case of furtherrefinements.

Figure 4.10. = 0.5 : Pressure (left) and velocity (right)

Figure 4.11. = 0.01 : Pressure (left) and velocity (right)

Figure 4.12. = 0 : Pressure (left) and velocity (right), (substantial drag)

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The following pictures demonstrates the velocity using streamlines. In left, the

Euler solution ( = 0) has been presented while the N.S solution with small friction( = 0.005) and turbulent fluid is shown in right.

Figure 4.13. Euler solution (left) and N.S solution (right)

4.2.2 Rotating Ball

Topspin :

Plots of how the drag force acts in time for a Coarser mesh with 12544 nodes and afiner mesh with 103918 nodes has been presented. The drag force varies in a small,constant range in time which gives us the ability to compute a reasonable value forthe drag force and at some point in time the mean value of Fd will act regularly.In figure(4.15), the zoomed in plots are demonstrated in which it is noticeable thatthe length of range changes of Fd is very small. The plots of Fd against time hasbeen presented below,for the simulation of rotating sphere, for two different meshes(after one and four refinements).

Figure 4.14. Drag force (Fd) in time for the coarse mesh (left) and the finer mesh(right).

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Figure 4.15. Drag force at the end of time interval for the coarse mesh (left) and

the finer mesh (right)

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Computations

When a sphere rotates in addition to the drag force that we disscussed before,

another force named lift force (Fl) will produce. This requires us to investigate thisforce and to visualize the data received using the computational method in hand.The following plots shows the lift force in time for two different meshes with 12544and 103918 nodes (respectively the coarse mesh in the first, the finer mesh in fourthrefinements). Investigating the results it has shown that the lift force will act in theregular, certain way after a time interval has passed. Therefore the same thing thatwe mentioned for the drag force stands for the lift force as well and we are able tohave a meaningful mean value of the lift force in time.

Figure 4.16. Lift force in time for the coarse mesh (left) and the finer mesh (right)

Figure 4.17. Lift force at the end of time interval for the coarse mesh (left) and thefiner mesh (right)

Plots of drag force and lift force has been presented in the following for differentmesh refinements. The drag force will become less oscillatory as the mesh is refinedwhereas the lift force becomes more oscillatory with mesh refinement. However withlift forces we observe that even though for higher mesh refinements we have more

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oscillations about the force axis, we still remain in a finite range which gives the

ability to easily take the mean value of Fl as a meaningful, computable value.

Figure 4.18. Drag force (Fd) vs. time in different mesh refinements (left) and liftforces (Fl) vs. time as we refine the mesh (right)

#ref # nodes cl cd fl fditr00 12544 -0.4109 0.8796 -0.1614 0.3482itr01 14481 -0.4264 0.8699 -0.1675 0.3416itr02 16834 -0.4517 0.8561 -0.1774 0.3362itr03 19617 -0.4091 0.8346 -0.1607 0.3278itr04 22755 -0.4023 0.7964 -0.1580 0.3127itr05 26407 -0.3898 0.7742 -0.1531 0.3040itr06 30607 -0.4008 0.7893 -0.1574 0.3100higher 103918 -0.4517 0.8561 -0.1835 0.2954

Table 4.1. Computational results of mean value lift coefficient (cl), drag coefficient

(cd),lift force (fl), and drag force (fd) for different mesh refitments

In the following plots the behavior of the drag and lift force with respect to log10number of mesh points has been presented. Although it is not possible to rule aaccurate prediction on how the function will act with further mesh refinement withthe few experiments, it is obvious that the drag force is decreasing intensely whilethe lift force seems to act more monotonous with the mesh refinement.

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Figure 4.19. Drag force vs. log10

of mesh points (left) and lift force vs. log10

of

mesh p oints (right)

Here some results from topspin generated by the solver is demonstrated usingParaview as visualizational tool. The results belong to rotational case and with103918 number of mesh points. The first two screen shots exhibitad the wake andseparation points. We can see that for the rotational sphere the separation points

are not summetric and as shown in the screen shots, we have a delay separation inleft in case of topspin. In the second pair of screen shots, we can see the results forpressure and velocity. The flow is unsymmetrical and lift force is produced becauseof the delay in separation. The low pressure in left will cause the sphere to moveleft.

Figure 4.20. Wake (left) and separation (right) in x2x3-plan

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Figure 4.21. Pressure solution (left), velocity solution (right)

The following figures show nice results of turbulence formed behind the movingsphere. The shows the turbulent and wake using streamlines and glyph as toolsavailable for users in Paraview.

Figure 4.22. Separation and wake in left (with glyphs) and turbulence in right(with streamlines), in x2x3-plan

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Figure 4.23. Screen shots of turbulence generated behind the sphere

The following pictures are the velocity solutions for rotating sphere, solved fora mesh with 98766 number of mesh points. Three samples of velocity solution hasbeen demonstrated in screen shots while the flow is developing in time.

Figure 4.24. Velocity solutions in different tme steps

For a mesh with 98766 number of mesh points, velocity and pressure solutionsas well as dual solutions underlying the adaptive mesh refinement algorithm usinga posteriori error estimation of the force on the sphere has been viewed below.

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Figure 4.25. Solutions of pressure (upper left) and velocity (upper right) and Dualsolutions for pressure (lower left) and velocity (lower right).

In the following pictures, we are able to see the mesh refinement for two differentmeshes. The blue edges corresponding to edges from the initial mesh (with 2404mesh points)and the white edges are the edges added in mesh refinement procedure(mesh with 3460 nodes).The left figure is in x1x3-plan and the right figure is inx1x2-plan.

Figure 4.26. Mesh refinement

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Backspin :

The following solutions are extracted from solving the problem assuming the sphereto rotate in the opposite direction of the topspin case. The results are produced byimplementing the model and boundary conditions to the mesh with 103918 nodes.In the first pair of screen shots, the wake and separation ponts are presented inx2x3-plan. It is noticable that there is a delay in the separation in right side. Thesecond pair of screen shots, show the pressure and velocity. The unsymmetrical flowwill produce a lift force. The low pressure in right will cause the sphere to moveright.

Figure 4.27. Wake (left) and separation (right)

Figure 4.28. Pressure solution (left), velocity solution (right)

# nodes cl cd fl fd103918 -0.4623 0.7508 -0.0713 0.2949

Table 4.2. Computational results of mean value lift coefficient(cl), dragcoefficient(cd),lift force(fl), and drag force(fd) for different mesh refitments

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4.2.3 Trajectory

(i)Experiments for a sphere with diameter = 1

Trajectory of sphere with the initial velocity U0 = (0, 1, 1).In the following figure the trajectory for the topspin has landed sooner with

compare to the other two. The reason is the backward lift force which was produceddue to the spin of the sphere has added a component vector in the direction of thegravity force; Therefore, we have an extra component force which will force thesphere to land sooner. With the same reason we can explain why the trajectory ofbackspin landed later. The vertical component of the lift force is opposite to the

direction of gravity force and therefore resists to reach to the ground sooner. Forthe still ball the forces are the ball are gravity and velocity. The non-rotating ballwill move in the direction of the horizontal velocity component and az the velocitydecreases, the sphere can move forward until the gravity force is larger than thevertical component of velocity. This is when the sphere will go downwards.

Figure 4.29. U0 = (0, 1,1), D = 1

Trajectory of sphere with the initial velocity U0 = (0, 50, 1).In the following figure both the initial angle and speed in which the sphere has

thrown increased compare to the previous figure. The trajectory for the topspin haslanded later. This is because the initial velocity is high the aerodynamic forces aremore important with respect to gravity and the sphere will travel further in the air.

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Figure 4.30. U0 = (0, 50,1), D = 1

Trajectory of sphere with the initial velocity U0 = (0, 300, 10).In the following figure the initial angle has not changed much with compare to

the previous figure, just a little decrease. However the initial speed has increaseda lot. The same behavior will happen with compare to the previous figure. The

difference here is that due to the higher initial velocity, gravity is even less importantthan the previous figure.

Figure 4.31. U0 = (0, 300,10), D = 1

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Trajectory of sphere with the initial velocity U0 = (0, 20, 1).

The same happens in the following figure, as in previous figures.

Figure 4.32. U0 = (0, 20,1), D = 1

(ii)Experiments for a sphere with diameter = 0.1

Trajectory of sphere with the initial velocity U0 = (0, 1, 1).In this figure the sphere initial velocity is the same as the experiment in figure

(4.31). The trajectory shows that the landing point is farther from the initialposition when using a sphere with smaller diameter in compare to the trajectory ofthe sphere with a larger diameter. Also, the plot has shown that the lift force withthe chosen initial conditions will not have a large effect on the sphere. As seen inthe plot the three trajectories travel very close together until they hit the groundin close positions together; This is shown in right plot.

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Figure 4.33. U0

= (0, 1,

1), D = 0.1

Trajectory of sphere with the initial velocity U0 = (0, 30, 5).The folowing figure is the same as the previuos one. The difference is that the

area of the sphere is smaller which will cause the aerodynamic forces to be lessimportant with respect to gravity and therefore it lands earlier.

Figure 4.34. U0 = (0, 30,5), D = 0.1

Trajectory of sphere with the initial velocity U0 = (0, 10, 5).In the following figure the interesting part is what happens at the top of the

hill. At the top of the hill the vertical component of the lift force is upwards and isthe most effective component here (since the horizontal component is almost zero).Comparing to the non-rotating sphere trajectory, we can conclude that with theinitial conditions of the problem, the vertical component is larger than the gravity

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Computations

force; Therefore, it will cause a lift at the top of the trajectory. This will make the

sphere to travel higher in the air. The same happens in the case of the top spin.The two forces, gravity and the vertical component of the lift force, will force thesphere to travel downward.

Figure 4.35. U0 = (0, 10,5), D = 0.1

Trajectory of sphere with the initial velocity U0 = (0, 50, 20).The trajectory of the backspin will go rightwards as the cause of resultants of

forces of gravity, velocity, and the lift force. The gravity force will act weaker asthere is a lift force upwards which will cause the sphere to travel further upwards.For the trajectory of the topspin, we think that probably there has been an errorin "offline" approach since the topspin has moved backwards.

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Figure 4.36. U0 = (0, 50,20), D = 0.1

Trajectory of sphere with the initial velocity U0 = (0, 4, 6).This trajectory is an example in which the trajectories in all three different

cases will travel very close to each other. However, in the right plot we can seethat they landed on the ground at different places and the lift force has effected the

trajectories. With initial conditions in this case the effect of the force seems to belesser than other problems (initial conditions).

Figure 4.37. U0 = (0, 4,6), D = 0.1

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Chapter 5

Summary

5.1 Conclusion

Using the G2 method, we were able to simulate the drag force of the non-rotatingsphere traveling in air. Also, we computed the drag and lift forces on the rotatingsphere. We have computed the separation, wake, flow velocity, pressure and tur-bulent flow that were formed behind the moving sphere. Using the offline solverthen we were able to plot the trajectories of some test cases with different initialconditions. The computations of trajectories appear qualitatively correct.

We noticed that the finer the mesh was, the computed drag and lift had moreoscillations in their graphs. In finer meshes, the drag coefficient decreases while thelift coefficient increases as the mesh refines.

The larger |u0| =

u0x + u0y + u

0z (u

0 being the initial velocity) was, the larger

the effects of aerodynamic forces (i.e. drag and lift) are compare to weight (W =mg).

Depending on the angle in which the sphere has thrown, it will differ whetherthe trajectory of topspin will land sooner than the trajectory of backspin or viceversa. This is depending on the effect and direction of the lift force on the globalcoordinate system.

5.2 Future work

In future work, we would like to apply gravity to the NSESolver in unicorn. Thisenables us to compute the trajectory in the solver more accurately. Also, we like tomodel as a function of velocity and even roughness of the sphere instead of a lineartime dependent function. The beta chosen here was to simplify the problem andwe can compute the flow around a sphere in a more realistic situation by choosing as an accurate function of velocity. Another change that we can make is to usethe solver to compute the flow for a specific sports ball with specific properties.

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Bibliography

[1] K. Horigan C. J. Pregnalato, M . C. Thompson. Flow transition in the wakeof a rotating sphere. Technical report, Advanced Computing Center, RIKEN

and NEC Informatec Systems, Ltd and Communication Media Research, NECCorporation, 2001.

[2] K. Horigan C. J. Pregnalato, M. C. Thompson. Flow patterns in the wakeof a streamwise-rotating sphere. Technical report, Advanced Computing Cen-ter, RIKEN and NEC Informatec Systems, Ltd and Communication MediaResearch, NEC Corporation, 2002.

[3] J. Hoffman. Computation of mean drag for bluff body problems using adaptivedns/les. SIAM J. Sci. Comput., 27(1):1841207, 2005.

[4] J. Hoffman. Efficient computation of mean drag for subcritical flow past acircular cylinder using general galerkin g2, accepted for int. J. Numer. Meth.Fluids,, 2005.

[5] J. Hoffman. Computation of turbulent flow past bluff bodies using adaptivegeneral galerkin methods: Drag crisis and turbulent euler solutions. Computa-tional Mechanics, 38, 2006.

[6] J. Hoffman. Simulating drag crisis for sphere using skin friction boundary con-ditions. European Conference on Computational Fluid Dynamics, ECCOMASCFD, May 2006.

[7] J. Hoffman and C. Johnson. Computational Turbulent Incompressible Flow,volume 4. Springer, 2006.

[8] J. Hoffman and C. Johnson. A new approach to computational turbulencemodeling. Comput. Methods Appl. Mech. Engrg., 195:28652880, 2006.

[9] J. Hoffman and C. Johnson. Finally : Resolution of dalemberts paradox. J.Math. Fluid Mech., May 2008.

[10] Acousto-Scan John Dunlop. Free flight aerodynamics of sports balls. 2003.

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Summary

[11] Shun Doi Ryutaro Himeno, Hideki Matsumoto. A numerical analysis of flows

around a rotating baseball. Technical Report 30, Department of MechanicalEngineering, Monash University, Australia, 2000.

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TRITA-CSC-E 2009:097

ISRN-KTH/CSC/E--09/097--SE

ISSN-1653-5715

simulation of in compressible flow around rotating sphere

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