main

Flow past a sphere at low Reynolds numbers

A Thesis Submittedin Partial Fulfillment of the Requirements

for the Degree of

Master of Technology

by

Navrose

Department of Aerospace Engineering

Indian Institute of Technology Kanpur

India

June, 2010

Certificate

Certified that the work contained in the thesis entitled “Flow past a sphere at low

Reynolds numbers” by Navrose, has been carried out under my supervision and it

has not been submitted elsewhere for a degree.

Dr. Sanjay Mittal

Professor

Department of Aerospace Engineering

Indian Institute of Technology, Kanpur

India-208016.

June, 2010

To Genelia

Abstract

Flow past smooth sphere placed in a uniform stream is investigated via three- dimen-

sional calculations. A stabilized finite element method is utilized to solve the incom-

pressible Navier-Stokes equations in the primitive variables formulation. Gambit is used

to generate the mesh aound the sphere. The elements of the mesh are a combination

of 4-noded tetrahedral, 5-noded pyramidal and 6-noded prosm elements. Computations

are carried out for Reynolds number in the range 100 to 1000. The flow features and the

aerodynamic coefficients are found to compare well with existing results. Many features

of the flow such as onset of hairpin vortex shedding in the wake is captured well by

the computations. A fine mesh is used to compute the Re=300 and Re=400 flow. The

computations are carried out on a distributed memory parallel computer. Speedup is

estimated for the two meshes. Preliminary results on flow past circular cylinder with

trip is presented at high Reynolds number. The trip is found to play a major role in

transition of boundary from a laminar to turbulent state.

iv

Acknowledgments

HEllo

v

Contents

Certificate ii

Abstract iv

Acknowledgments v

Contents vi

List of Figures viii

Nomenclature xi

1 Introduction 1

1.1 Flow past smooth sphere . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Flow past sphere with trip inclined at an angle . . . . . . . . . . . . . 5

2 Governing Equations and finite element formulation 8

2.1 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Stabilization parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 13

vi

CONTENTS vii

3 Results and discussions 15

3.1 Flow past a circular cylinder with single and two trips . . . . . . . . . 15

3.2 Flow past sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 Scalability study . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Flow past sphere with trip . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Conclusions 37

References 39

List of Figures

1.1 Achenbach sketch of flow past a sphere at Re = 400. . . . . . . . . 5

1.2 Typical 2D grid used by Mittal and Najjar [1]. 3D mesh is generated

by by rotating this mesh about x-axis resulting in spheroid domain. 6

1.3 Flow past a sphere: variation of Strouhal number with Reynolds num-

ber. The figure has been taken from Mittal et al.[1]. . . . . . . . . 6

1.4 Flow past a sphere: vortex shedding in the sphere wake at differ-

ent range of Reynolds numbers. The picture has been taken from

Sakamoto and Haniu [2]. . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Schematic of the problem set-up: boundary conditions. . . . . . . 10

3.1 Variation of aerodynamic coefficients with Re on smooth cylinder,

cylinder with one trip and cylinder with two trips. The computational

data is taken from Behara et. al. . . . . . . . . . . . . . . . . . . . 21

3.2 Isometric view and Cross-section of the mesh M2. Mesh M1 was made

with same dimensions but with lower number of elements . . . . . 22

3.3 Flow past a cylinder with trip : Re=150000 (top figure) shows more

suction on the side of the trip and Re=400000 (bottom figure) shows

opposite behaviour. X-axis represents the angular positions on the

surface of the cylinder. θ = 0 represents the forward stagnation point. 23

viii

LIST OF FIGURES ix

3.4 Comparison of CD and CL obtained from 2D and 3D computation of

flow past cylinder with trip at 55 . The computational data is taken

from Behara et. al. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Conversion of four noded, five noded and six noded elements into Hex-

ahedral elements for finite element computations. . . . . . . . . . . 25

3.6 Flow past a sphere : Time averaged drag coefficient as a function of

Reynolds number. The results from Almedeij and Wieselsberger is also

included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.7 Flow past a sphere : Time averaged lift coefficient as a function of

Reynolds number. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.8 Flow past a sphere : Time averaged side force coefficient as a function

of Reynolds number. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.9 Flow past a sphere : Energy spectra at different Reynolds number. No

frequency is observed for Re=100 and Re=250. . . . . . . . . . . . 28

3.10 Flow past a sphere : Energy spectra at different Reynolds number. No

frequency is observed for Re=100 and Re=250. . . . . . . . . . . . 29

3.11 Flow past a sphere : iso-surfaces for the stream-wise component of

velocity (left column)and stream-wise component of vorticity (right

column) in a sectional view. ωx = -0.1 in blue and ωx = 0.1 in red

at different Reynolds number. At Re =100 no isovorticity surfaces are

seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.12 Flow past a sphere : iso-surfaces for the stream-wise component of

velocity (left column)and stream-wise component of vorticity (right

column) in a sectional view. ωx = -0.1 in blue and ωx = 0.1 in red

at different Reynolds number. At Re =100 no isovorticity surfaces are

seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.13 Flow past a sphere : time history of aerodynamic coefficients(CD and

Cl) at different Reynolds number. Beyond Re =300 oscillations are

seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

LIST OF FIGURES x

3.14 Flow past a sphere : time history of aerodynamic coefficients(CD and

Cl) at different Reynolds number. Beyond Re =300 oscillations are

seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.15 Flow past a sphere: stream-wise vorticity isosurfaces for one period of

vortex shedding with blue representing ωx = −0.1 and red representing

ωx = +0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.16 Speedup S obtained on various number of processors with meshes M1

and M2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.17 Flow past a sphere : Comparison of instantaneous stream-wise com-

ponent of velocity after t=25s for mesh M1 and M2 at Re=400.The

figure on the top is for M1 and lower one is for M2. The coarse mesh

has run for 200 time steps and fine mesh for 2500 time steps. . . . 36

Nomenclature

Ω Spatial domain

Γ Boundary of the spatial domain (Ω)

nsd Number of spatial dimensions

Rnsd Real space with nsd dimensions

ik Orthonormal basis in the Cartesian coordinate system

x Spatial coordinate

t temporal coordinate

u Velocity vector

ui Cartesian components of velocity vector

un, ut Normal and tangential components of velocity vector

p Pressure

CL Lift Coefficient

CD Drag Coefficient

CS Side force Coefficient

Re Reynolds number

Lc Characteristic length of geometry

T Viscous stress tensor

St Strouhal number

Sh Finite element trial function space

Vh Finite element weighting function space

H1h(Ω) Finite dimensional function space over Ω

Wh Weighting function

τ Stabilization coefficient tensor for SUPG

Jik Components of the Jacobian transformation matrix

j Determinant of the Jacobian transformation matrix

Nomenclature xii

ndof Number of degrees of freedom

nel Number of elements

nnp Number of global nodes

nen Number of elemental nodes

h Element length scale

Na Interpolation function associated with node a

η, ηgiSet of global nodes, Dirichlet nodes for ith degree of freedom

2–D Two dimensional

3–D Three dimensional

N-S Navier Stokes

Chapter 1

Introduction

Flow past bluff bodies is of relevance in various fields such as ocean, civil, power, nu-

clear, aerospace engineering. At sufficiently small Reynolds numbers, these flows are

known to be attached and steady. At higher Reynolds numbers, the flow becomes sep-

arated,unsteady and finally turbulent. The nature of these transitions have important

consequences in practical applications. The flow of a viscous fluid past a stationary iso-

lated sphere and cylinder may be considered as a simplified case of bluff body flows. For

example, the phenomenon of drag crisis in circular cylinders, which has been extensively

researched upon, is also observed for other bluff body flows. This is attributed to the

transition of boundary layer from laminar to turbulent. This transition, as mentioned,

significantly affects the aerodynamic forces experienced by the body. The present nu-

merical simulation is intended to investigate the aerodynamic forces experienced by a

sphere with trip positioned at an angle to the incoming flow. To the best knowledge of

this author, no computational work has been reported for such flows. 2D computations

for flow past cylinder with trips is presented to gain insight into the effect of trip on

aerodynamic coefficients. 2D and 3D computational results from Suresh et al. [3] are

also presented. Numerical simulations are then carried out for flow past sphere at low

1

1.1 Flow past smooth sphere 2

Reynolds numbers. A hybrid mesh consisting of different elements is used.

1.1 Flow past smooth sphere

Sphere leaves a prototypical axisymmetric wake, frequently encountered in many engi-

neering applications. In laminar regime, the sphere wake undergoes various transitions

as the Reynolds number, Re, is increased. These transitional regimes serve as a check-

point for numerical computations. These regimes are mainly based on flow structures,

aerodynamic coefficients, and Strouhal number. The general trends in these regimes,

specified below, have been confirmed by many researchers.

Taneda’s flow visualization experiment established that when Reynolds number is

over 130, faint periodic pulsative motion with very long period occurs forming a wavelike

wake. This wave motion in the wake is observed until Reynolds number 300 is reached.

Beyond this Reynolds number, the axisymmetric pattern of the flow breaks down and

hairpin shaped vortices are shed periodically with regularity in strength. Particularly

interesting throughout the literature is that these vortices are all shed with the same

orientation , forming a ladder like chain of overlapping loops (Figure 1.1) This shedding

of vortices in a sphere wake is substantially different from that in wake of a cylinder.

Margaravey and co-workers initiated a detailed investigation of the structure of these

vortices to understand the mechanism behind these instabilities. Achenbach [4] reported

that this phenomenon of vortex shedding is observed when Re=400 is reached. Guschin

et al. [5], did direct numerical simulation of flow around the sphere and reported that

the hairpin shaped vortices separate from only one half of the sphere. Hence time

averaged lift/side-force and torque coefficients are not equal to zero. Sakamoto and

Haniu [2] conducted experiments at Re= 400. They reported that the waveform of the

fluctuating velocity based on shedding of hairpin vortices, become irregular at this Re.

2


The reason cited is the imbalanced supply,storage, and emission of energy within the

vortex formation region. They also observed that the non-dimensional frequency, F,

of the vortices from the sphere is distributed on different straight lines in regular and

irregular mode with Reynolds number. Similar results for cylinder have been confirmed

by Roshko[6]. The non dimensional frequency F, is given by

F =f ∗ d2

ν(1.1.1)

Here f is the frequency of hairpin-shaped vortex shedding, d is the diameter of the

sphere and ν is the viscosity of the medium. In the Reynolds number range 480-650,

the orientation of the vortex varies significantly from cycle to cycle. Sakamoto and

Haniu have suggested a change in the shedding point of hairpin shaped vortices for this

observation. Accordingly, as Taneda(1978) has pointed out, the vortices are shed as if

a facet including the vortices are rotating slowly and irregularly about an axis through

the centre of the sphere in the direction of undisturbed flow. Beyond Reynolds number

800, experiments have shown that two instability modes, a high mode and a low mode,

co-exist[4][2]. Figure 1.3 shows the variation of Strouhal number with Reynolds number.

The low mode is caused by the progressive wave motion of the wake with alternate

fluctuations. The high mode is attributed to periodic shedding of vortex sheet from the

surface of the sphere. As the Reynolds number is further increased, the complexity of

vortex shedding in the wake increases. Multiple peaks corresponding to several dominant

frequencies in the wake are observed in the energy spectra by Mittal et al.[1]. They

report non-linear interaction between the vortex shedding frequency and other lower

frequencies. So, although much research has been done to study the structure of wake

in this region, little consensus exists[1]. A schematic showing different flow regimes is

shown in Figure 1.4

3


Generation of finite element mesh around sphere has always been a challenging

task in computational fluid dynamics. Two methods of generating the mesh have been

reported in literature. In the first method a 2-D mesh is generated around a cross-section

of the sphere as shown in Figure 1.2. 3-D mesh is then obtained by a rotation of the 2-D

mesh about the axis of the sphere, represented by the x-axis in the figure. This method

almost always results in structured meshes in a spherical outer domain. Hence, increasing

the resolution of mesh to capture the dynamics of flow in the boundary layer requires

large computational resources. The other method involves a hybrid mesh that consists

of a structured part close to the sphere and an unstructured mesh in the remaining

of the domain Very few research groups have adopted this method[7]. In the present

calculation, Gambit software [mesh generation too from ANSYS] is used to generate a

hybrid mesh around the sphere. This mesh is then used to simulate the flow past a

sphere. The resultant force experienced by the sphere can be decomposed along three

perpendicular directions: drag in the stream-wise direction and lift and side force in the

plane normal to the free-stream direction. The related non-dimensional coefficients are

denoted by CD, CL and CS.

In the current study, we simulate incompressible flow around a smooth sphere in

the discussed flow regimes. The objective is to compare the solution obtained from the

numerical computations using hybrid mesh to existing experimental and computational

results. The simulations are carried out for RE = 100, 250, 300, 400, 500, 700, 1000 and

1400. The results for the Re=400 flow are compared with results computed on a much

finer mesh. Table 3.1 compares the nodes and elements in the two meshes. Scalability

is also studied by implementing the stabilised finite element formulation on a parallel

computer for the two meshes mentioned above. The Linux cluster is a 12 node, 32-bit

precision, parallel computing machine loaded with MPI libraries. Each node has eight

dual core xeon processors with clock speed 2.33 GHz and a main memory unit of 8 GB

4

1.2 Flow past sphere with trip inclined at an angle 5

Figure 1.1: Achenbach sketch of flow past a sphere at Re = 400.

RAM.

1.2 Flow past sphere with trip inclined at an angle

The motivation behind studying flow past a sphere with trip is to study the aerodynam-

ics of a cricket ball. Many experimental results have been reported by researchers by

mounting the cricket ball in a wind tunnel[8][9]. But the mount interferes with the flow

and affects the dynamics of flow around the ball. A computational environment provides

the opportunity to study the flow past a ball without the interference/disturbance of the

mount. Hence 3D computations can give a better insight into the requisite aerodynam-

ics. A hybrid mesh is made around a sphere with trip, similar to the smooth sphere as

described earlier. The trip height is approximately equal to the seam height on a cricket

ball. The mesh in this case is finer than the one used for a smooth sphere and contains

13,729,800 nodes and 18,024,691 elements. This is done to have adequate number of

nodes around the trip to accurately capture the flow dynamics.

5


Figure 1.2: Typical 2D grid used by Mittal and Najjar [1]. 3D mesh is generated by byrotating this mesh about x-axis resulting in spheroid domain.

Figure 1.3: Flow past a sphere: variation of Strouhal number with Reynolds number.The figure has been taken from Mittal et al.[1].

6


Figure 1.4: Flow past a sphere: vortex shedding in the sphere wake at different range ofReynolds numbers. The picture has been taken from Sakamoto and Haniu [2].

7

Chapter 2

Governing Equations and finite

element formulation

In this chapter the governing equations for the viscous incompressible flow in three

dimensions is presented. This is followed by the finite element discretization that is used

to solve these equations.

2.1 The Navier-Stokes equations

Let Ω ⊂ Rnsd and (0, T ) be the spatial and temporal domains, respectively, where nsd is

the number of space dimensions, and let Γ denote the boundary of Ω. The spatial and

temporal co-ordinates are denoted by x and t. The Navier-Stokes equations governing

incompressible flow are

ρ

(

∂u

∂t+ u.∇u

)

− ∇.σ = 0 on Ω × (0, T ) (2.1.1)

∇.u = 0 on Ω × (0, T ) (2.1.2)

8

2.1 The Navier-Stokes equations 9

Here ρ, u and σ are the density, velocity and stress tensor, respectively. The stress

tensor is written as the sum of its isotropic and deviatric parts:

σ = −pI + T , T = 2µǫ(u), ǫ(u) =1

2(∇u + (∇u)T )

where p, I and µ are the pressure, identity tensor and dynamic viscosity, respectively.

The boundary conditions are either on the flow velocity or stress. Both, Dirichlet and

Neumann type boundary conditions are accounted for:

u = g on Γg (2.1.3)

n.σ = h on Γh (2.1.4)

where, n is the unit normal vector on the boundary Γ. Here, Γg and Γh are subsets of the

boundary Γ. More details on the boundary conditions are given in Figure 2.1. ΓU , ΓD

and ΓS represent the upstream, downstream and lateral boundaries, respectively. The

surface of the body is represented by ΓB.

The initial condition on the velocity is specified on Ω:

u(x, 0) = u0 on Ω (2.1.5)

where u0 is divergence free.

The drag and lift force coefficients, (Cd, Cl), on the body are calculated using the

following expression:

(Cd , Cl) =2

ρU2S

∫

ΓB

σn dΓ (2.1.6)

9

2.2 Finite element formulation 10

Figure 2.1: Schematic of the problem set-up: boundary conditions.

2.2 Finite element formulation

The finite element method has been a popular method for solving partial differential

equations. Among the characteristics of this method that has led to its widespread

use is its ease in modelling complex geometries. The finite element method is a piece-

wise approximation in which the approximating function is formed by connecting simple

functions, each defined over a small region, namely the element. The approximations are

based upon the Galerkin formulation of the method of weighted residuals. The Galerkin

formulation leads to a coupled system of linear algebraic equations. This method has

found great success in applications to problems of structural mechanics, heat conduc-

tion and other related problems governed by diffusion type equations. When applied

to problems governed by self-adjoint elliptic or parabolic partial differential equations,

this method leads to symmetric matrices. The difference between the solution of finite

element method and the exact solution in this case, is minimized with respect to an

energy norm [10]. The Galerkin method is seen to be optimal for such problems.

10


Galerkin methods do not enjoy the same advantages when applied to fluid flow

problems domination by convection. The problem is that convection operators are non-

symmetric, and therefore the best approximation property in the energy-norm is lost.

The Galerkin finite element method when applied to convection dominated flows suffers

from two major disadvantages.

1. The solution obtained from this method are often characterized by spurious node

to node oscillations. The Galerkin finite element discretization when used with

linear approximations coincides with that of second-order central differences. The

reason for this node to node oscillation is that the formulation introduces a negative

numerical diffusion.

2. The second disadvantage is seen primarily while using mixed methods. The func-

tion spaces representing the velocity and pressure fields need to be compatible.

They need to satisfy the inf-sup or the LBB condition of Ladyzhenskaya [11],

Babuska [12] and Brezzi [13]. The inf-sup condition states that the velocity and

pressure spaces cannot be chosen arbitrarily, a link between them need to exist. In

the absence of the necessary link, the pressure field is likely to exhibit an oscilla-

tory behaviour. Most commonly found compatible functions spaces, which involve

equal in order interpolations for velocity and pressure, do not satisfy the inf-sup

condition.

The above disadvantages could be remedied by either using a significantly refined

mesh and using function spaces that satisfy the inf-sup condition or by adding numerical

diffusion into the formulation and choosing function spaces that circumvent the inf-

sup condition. The former leads to significant increase in computational effort and is

therefore rarely followed. The later leads to what is called the stabilized formulations.

Stabilized formulations have been popular while dealing with convection dominated flow

11


problems. The Streamline Upwind Petrov-Galerkin(SUPG) technique introduced by

Brooke and Hughes [14] introduces numerical diffusion which acts only in the direction

of the flow. This is achieved by adding a streamline upwind perturbation term to the

weighing function. Since, the space of weighing functions and interpolation functions are

no more coincident, this is a Petrov-Galerkin formulation. A technique to circumvent

the inf-sup condition is given for Stokes flow by Hughes [15]. When this technique is

extended to finite Reynolds number flows it leads to the Pressure Stabilizing Petrov-

Galerkin formulation(PSPG) of Tezduyar et al. [16]. This technique enables the usage

of equal in order interpolation functions for pressure and velocity without generating

oscillations in the pressure field.

Consider a finite element discretization of Ω into subdomain Ωe,e = 1, . . . , nel,

where nel is the number of elements. We define:

H1h(Ω) = φh|φh ∈ C0(Ω), φh|Ωe ∈ P 1, e = 1, 2, . . . , nel (2.2.7)

with P 1 representing first-order polynomials. The trial and test function spaces are

defined as:

Shu = uh|uh ∈ (H1h)nsd, uh = gh on Γg (2.2.8)

Vhu = wh|wh ∈ (H1h)nsd, wh = 0 on Γg (2.2.9)

Shp = Vh

p = qh|qh ∈ H1h (2.2.10)

where nsd is the number of space dimensions.

The stabilized finite element formulation of Equations (2.1.1) and (2.1.2) is written

12

2.3 Stabilization parameter 13

as follows: find uh ∈ Shu and ph ∈ Sh

p such that ∀wh ∈ Vhu , qh ∈ Vh

p

∫

Ω

wh.ρ

(

∂uh

∂t+ uh.∇uh − f

)

dΩ +

∫

Ω

ǫ(wh : σ(ph, uh)) dΩ

+

∫

Ω

qh∇.uh dΩ +

nel∑

e=1

∫

Ωe

1

ρ(τSUPGρuh.∇wh + τPSPG∇qh).

[

ρ

(

∂uh

∂t+ u.∇u − f

)

− ∇.σ

]

dΩe

+

nel∑

e=1

∫

Ωe

τLSIC∇.whρ∇.uh dΩe =

∫

Γh

wh.hh dΓ. (2.2.11)

The first three terms and the right-hand side of Equation (2.2.11) constitute the Galerkin

formulation of the problem. The terms involving the element level integrals are the

stabilization terms added to the basic Galerkin formulation to enhance its numerical

stability. The term with coefficient τLSIC is also a stabilization term based on the least

squares of the incompressibility constraint and is found to be useful for large Reynolds

number flows. The generalized trapezoidal rule is used for time discretization of uh.

This is given by:

uh = αui + (1 − α)ui+1, (2.2.12)

∂uh

∂t=

ui+1 − ui

∆t(2.2.13)

where 0 ≤ α ≤ 1, and ui represents the velocity field at time step i.

2.3 Stabilization parameter

The coefficients τSUPG and τPSPG in Equation (2.2.11) are given by:

τSUPG = τPSPG =

[

(

2||uh||

h

)2

+

(

12ν

h2

)2]

(2.3.14)

13

2.3 Stabilization parameter 14

where h is the element length. In the present work, h is taken as the minimum length

between two vertices in an element. The coefficient τLSIC is defined as

τLSIC =

[

(

2

h||uh||2

)2

+

(

12ν

h2||uh||2

)

]

−1/2

(2.3.15)

More details on the formulation can be found in Tezduyar et al. [16].

14

Chapter 3

Results and discussions

3.1 Flow past a circular cylinder with single and

two trips

Figure 3.1 shows the variation of Cd and Cl with Re for a smooth cylinder, cylinder

with one trip at 55 and cylinder with two trips at +/- 80 . The presence of roughness

element leads to lowering of critical Reynolds number for the onset of transition. In the

case of single trip, asymmetric flow and asymmetric suction exists on two halves of the

cylinder. This asymmetry is shown in Figure 3.3. At Re=150000, the boundary layer on

rough side is turbulent while on the other side it is laminar. Owing to the higher suction

by turbulent boundary layer the cylinder experiences a finite force towards rough side

which is otherwise zero for smooth sphere. At Re=400000, both the sides have turbulent

boundary layer but due to the presence of trip, the side has lower suction than the other

side, leading to reversal in the direction of force. This phenomenon as described later,

plays an important role in governing the kinematics of rough sphere moving in fluid.

Unlike the smooth cylinder, the drag crisis in this case happens in two stages. This is

due to the asymmetric transition on two halves of the cylinder.

15

3.2 Flow past sphere 16

Qualitatively, the flow over cylinder with trip at +/-80 is similar to flow over

smooth cylinder. As observed in Figure 3.1 the plot of Cd for cylinder with trip is

shifted from smooth cylinder. The general trend is similar in the two cases. Computa-

tions are currently going on to investigate the hysteresis in the variation of aerodynamic

coefficients with Re. The Reynolds number of a cricket ball decreases continuously in

flight. Hence an understanding of hysteresis in single trip is also essential to predict,

the kinematics of a cricket ball. Computations are also being carried out to observe

significant change in flow properties, if any, for the trips placed at 80 and −78 .

Drag coefficient and lift coefficient of a 3-D cylinder with trip at 55 , available from

Suresh [CFD lab, IIT Kanpur], is compared with 2D data for single trip over a cylinder.

Comparison is shown in Figure 3.4. In 3D computations also drag crisis and lift reversal

are observed. These results are instrumental in understanding the nature of flow past

sphere with trip.

3.2 Flow past sphere

3.2.1 Mesh Generation

The meshes employed in the current computations are generated using Gambit. Two

meshes are made with different number of elements. The number of nodes and elements

are shown in Table 3.1. The sphere resides in a computational domain whose outer

boundary is a hexahedron. The meshes consists of structured and unstructured part as

shown in Figure 3.2. The structured part consists of 6-node wedge shape elements. This

part is enclosed in a shell around the sphere and is adequately refined to capture the

dynamics of flow inside the boundary layer. In the remaining part of the domain, mesh

is unstructured. It consists of 4-noded tetrahedral, 5-noded hexahedral and 6-noded

wedge elements. For both meshes, in the vicinity of the sphere and in the wake region,

16


mesh is finer as compared to the rest of domain. This is done to give requisite resolution

for the flow in the domain. The region near the sphere in Figure 3.2 contains 8,071,765

elements. The next coarser region contains 1,394,121 elements. Rest of the domain

contains 282.922 elements.

The present computations are done for hexahedral elements. Hence all the elements

in the mesh are converted into equivalent hexahedral elements. This is realised by

repeating some nodes and accurately numbering them to ensure proper connectivity.

Figure 3.5 shows a schematic of this conversion.

3.2.2 Numerical Simulations

Flows at various Reynolds number are simulated and compared with the flow visualiza-

tion and computational results from different researchers. The variation of drag coeffi-

cient with Reynolds number is shown in Figure 3.6. Also shown are the experimental

results from Almedeij [17] and Wieselsberger [18]. The difference in the values of CD

may be due to the effect of mounts in experimental results, which is otherwise absent

in our numerical computations. Because of the symmetrical structure of the sphere it

is expected that time averaged lift coefficient and side force coefficient is equal to zero.

The variation of lift coefficient CL shown in Figure 3.7, shows asymmetric lift distribu-

tion on the two halves of the sphere in Re=250-400. At other Reynolds numbers the

value is close to zero. The reason is attributed to separation of hairpin-shaped vortices

from one part of the sphere. A similar observation is reported by Mittal and Najjar.[1]

Figure 3.8 shows the variation of side force coefficient CS. To the best knowledge of this

author, asymmetric behaviour of CS for low Reynolds number has not been mentioned

in literature.

Shown in Figure 3.11 and Figure 3.12 are the contours of instantaneous stream-wise

component of velocity and stream-wise vorticity isosurfaces (ωx = 0.1). These pictures

17


give a glimpse of the wake structure at various Reynolds number. No vortex shedding is

observed for Re = 100 and Re = 250. The flow structure is same for the two cases. A

look into the time histories of CD and CL in Figure 3.13 and Figure 3.14 also ascertains

steady flow at these Reynolds number by showing no oscillations. At Re = 300 hairpin

shaped vortices start shedding from the sphere. CD and CL show periodic oscillations

which is attributed to the vortex shedding at this Reynolds number. The amplitude of

the oscillation becomes constant after the flow has been stabilized. The effect of this

is the shedding of vortices with regularity in strength. Flow structures at Re=400 and

Re=500 is almost similar showing periodic vortex shedding. Figure 3.15 shows vorticity

isosurfaces along streamwise direction for one cycle of vortex shedding. The generation

of hairpin shaped vortices is captured in these figures. As Reynolds number is increased

beyond 400 irregularity in time histories of CD and CL is observed. This may be due to

irregularity in strength of vortices being shed. At Re=700 the wake structure is complex

but is laminar. Beyond Re = 700 turbulent structures start forming in the wake. CD and

CL also show irregular time histories. The above results agree well with data available

in literature.[2][1]

Figure 3.9 and 3.10 shows energy spectra at different Reynolds numbers. No peak

is observed for Re = 100 and Re = 250. Between Re = 300 and Re = 700 single

dominant frequency is seen. This dominant frequency corresponds to periodic shedding of

hairpin shaped vortices. Beyond Re=700 multiple peaks are obtained. These frequencies

interact non-linearly and result in complex evolution of the vortices in the wake These

observations also conform to earlier experimental results.

Comparison between simulation results of coarse mesh M1 and fine mesh M2 is

done at Re = 300 and Re = 400. The velocity contours show remarkable difference.

Vortex shedding is observed in M1 but no shedding is seen in M2 at Re = 400 Figure

3.17 shows the stream-wise velocity component at Re=400. This might be due to small

18


time step used for mesh M2. If the computations on fine mesh are allowed to run for

large number of time steps vortex shedding may also be seen in M2. The reason for this

conjecture is reflected in the values of CD for the two meshes at Re=300 and Re=400.

CD= 0.662 is obtained at Re=300 for mesh M2. This agrees well with earlier computed

value of CD= 0.657 for mesh M1. At Re=400 CD=0.573 for mesh M2 and CD=0.609 is

obtained. As Reynolds number increases, this difference is expected to be large because,

the coarse mesh cannot capture all the phenomenon in shear layer which the fine mesh

is expected to capture up to very high Reynolds number.

3.2.3 Scalability study

Scalability study is carried out for two finite element meshes M1 and M2 (details in Table

3.1). To investigate scalability speedup on different number of processors are computed.

Speedup is calculated as

S =(Tmin procs)(nmin procs)

Tn

(3.2.1)

Finite element code for the two meshes was run for one timestep only. The cases were

repeated to confirm the timings.

Figure 3.16 shows variation of speedup with number of processors. Ideal/linear

speedup for which the number of processors equal to number of node is also plotted. It

is observed that for both the meshes sublinear speedup is achieved. As the number of

equations in increased the speedup approaches towards being linear. Similar observa-

tion was made by Behara and Mittal.[19] Increasing processors beyond a certain number

(number depends on the mesh) results in very poor speedup. The reason for this be-

haviour is attributed to increase in the time spent for inter-processor communication as

number of processors are increased. Superlinearity was achieved by Behara and Mittal

[19] for large scale problems. They attributed this to cache related effects. In the present

19

3.3 Flow past sphere with trip 20

computations no superlinearity is seen because the problem size is relatively small.

3.3 Flow past sphere with trip

Numerical simulations are done for the flow past a sphere with trip. The results obtained

show numerical oscillations near the surface of the sphere. These oscillations are mainly

seen is the contours of pressure. At some nodes in the domain the magnitude of Cp goes

as high as 100. This is common in computations at Re=100, 300, 1000, 5000, 10000. Dif-

ferent procedures to eliminate the problem are proposed. Computations with two values

of k, dimension of Kyrlov subspace are carried out. Computations with impulsive start

are also done at high Reynolds numbers. The solution in all the cases show divergence.

Decreasing the value of time step below a certain value (which depends on Reynolds

number) improves convergence. However the iterations do not bring the residue down

to a permissible level in such cases. Also the Courant number based on minimum el-

ement length, time spent and local speed, show very high value (approx. 100)at some

elements. For a simple diffusion equation C∆t should be less than 1 for the solution to

be conditionally stable. This implies that the value of aerodynamic coefficients obtained

may not be correct. Three different mesh parameters (for the stabilization) based on

minimum length,maximum length and equivalent radius of sphere by equating volume

for each element are used. In all the three cases the trends as discussed above remains

the same. The problem might be due to the presence of a skewed element in the mesh

or very high aspect ratio elements in the boundary layer. A new mesh with appropriate

aspect ratio will be generated and then used in future computations. Other reason may

be that the continuity equation is not being solved accurately. For this we may have to

change the preconditioners that we are using currently.

20


Mesh Nodes Elements Number of equations nmin

M1 167087 569484 655776 1M2 5498703 9748808 21852445 3

Table 3.1: Details of finite element mesh used in the present study.nmin is the minimumnumber of processors below which the process stalls.

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

2.000

2.200

100 1000 10000 100000 1e+06

Cd

Re

smoothtrip at 55o

trips at +/-80o

-0.800

-0.600

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

100 1000 10000 100000 1e+06

Cl

Re

smoothtrip at 55o

trips at +/-80o

Figure 3.1: Variation of aerodynamic coefficients with Re on smooth cylinder, cylinderwith one trip and cylinder with two trips. The computational data is taken from Beharaet. al.

21


Figure 3.2: Flow past a cylinder with trip : Re=150000 (top figure) shows more suctionon the side of the trip and Re=400000 (bottom figure) shows opposite behaviour. X-axisrepresents the angular positions on the surface of the cylinder. θ = 0 represents theforward stagnation point.

22


Figure 3.3: Isometric view and Cross-section of the mesh M2. Mesh M1 was made withsame dimensions but with lower number of elements

23


0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

2.000

10000 100000 1e+06

Cd,

CD

Re

2D3D

-0.800

-0.600

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

10000 100000 1e+06

Cl,C

L

Re

2D3D

Figure 3.4: Comparison of CD and CL obtained from 2D and 3D computation of flowpast cylinder with trip at 55 . The computational data is taken from Behara et. al.

24


Figure 3.5: Conversion of four noded, five noded and six noded elements into Hexahedralelements for finite element computations.

25


0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

0 200 400 600 800 1000 1200 1400

CD

Re

Present CalculationWieselsberger

Almedeij

Figure 3.6: Flow past a sphere : Time averaged drag coefficient as a function of Reynoldsnumber. The results from Almedeij and Wieselsberger is also included.

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0 200 400 600 800 1000 1200 1400

CL

Re

Present Calculation

Figure 3.7: Flow past a sphere : Time averaged lift coefficient as a function of Reynoldsnumber.

26


-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

100 200 300 400 500 600 700

CS

Re

Present Calculation

Figure 3.8: Flow past a sphere : Time averaged side force coefficient as a function ofReynolds number.

27


0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0 0.2 0.4 0.6 0.8 1

Ek

St

Re = 300

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

0 0.2 0.4 0.6 0.8 1

Ek

St

Re = 400

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0 0.2 0.4 0.6 0.8 1

Ek

St

Re = 500

Figure 3.9: Flow past a sphere : Energy spectra at different Reynolds number. Nofrequency is observed for Re=100 and Re=250.

28


0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

0 0.2 0.4 0.6 0.8 1

Ek

St

Re = 700

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0 0.2 0.4 0.6 0.8 1

Ek

St

Re = 1000

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0 0.2 0.4 0.6 0.8 1

Ek

St

Re = 1400

Figure 3.10: Flow past a sphere : Energy spectra at different Reynolds number. Nofrequency is observed for Re=100 and Re=250.

29


Re = 100

Re = 250

Re = 300

Re = 400

Figure 3.11: Flow past a sphere : iso-surfaces for the stream-wise component of velocity(left column)and stream-wise component of vorticity (right column) in a sectional view.ωx = -0.1 in blue and ωx = 0.1 in red at different Reynolds number. At Re =100 noisovorticity surfaces are seen.

30


Re = 500

Re = 700

Re = 1000

Figure 3.12: Flow past a sphere : iso-surfaces for the stream-wise component of velocity(left column)and stream-wise component of vorticity (right column) in a sectional view.ωx = -0.1 in blue and ωx = 0.1 in red at different Reynolds number. At Re =100 noisovorticity surfaces are seen.

31


1.683

1.684

1.685

1.686

1.687

1.688

1.689

1.690

1.691

1.692

0 200 400 600 800 1000 1200 1400 1600 1800

CD

Time steps (dt = 0.1)

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0 200 400 600 800 1000 1200 1400 1600 1800

CL


Re = 100

0.697

0.698

0.698

0.698

0.698

0.698

0.699

0.699

0 200 400 600 800 1000 1200

CD


0.065

0.066

0.067

0.068

0.069

0.070

0.071

0.072

0.073

0.074

0.075

0 200 400 600 800 1000 1200

CL


Re = 250

0.654

0.655

0.656

0.657

0.658

0.659

0.660

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

CD


0.065

0.070

0.075

0.080

0.085

0.090

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

CL


Re = 300

0.590

0.595

0.600

0.605

0.610

0.615

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

CD


-0.080

-0.060

-0.040

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

CL


Re = 400

Figure 3.13: Flow past a sphere : time history of aerodynamic coefficients(CD and Cl)at different Reynolds number. Beyond Re =300 oscillations are seen.

32


0.565

0.570

0.575

0.580

0.585

0.590

0.595

0 200 400 600 800 1000 1200

CD


-0.120

-0.100

-0.080

-0.060

-0.040

-0.020

0.000

0.020

0.040

0.060

0.080

0.100

0 200 400 600 800 1000 1200

CL


Re = 500

0.515

0.520

0.525

0.530

0.535

0.540

0.545

0.550

0.555

0.560

0.565

0.570

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

CD


-0.150

-0.100

-0.050

0.000

0.050

0.100

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

CL


Re = 700

0.480

0.490

0.500

0.510

0.520

0.530

0.540

0.550

0.560

0.570

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

CD


-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

CL


Re = 1000

Figure 3.14: Flow past a sphere : time history of aerodynamic coefficients(CD and Cl)at different Reynolds number. Beyond Re =300 oscillations are seen.

33


Figure 3.15: Flow past a sphere: stream-wise vorticity isosurfaces for one period of vortexshedding with blue representing ωx = −0.1 and red representing ωx = +0.1.

34


0.0

20.0

40.0

60.0

80.0

100.0

0 20 40 60 80 100

S(s

peed

up)

n(number of processors)

mesh M1linear

mesh M2

Figure 3.16: Speedup S obtained on various number of processors with meshes M1 andM2.

35


Y X

Z

Figure 3.17: Flow past a sphere : Comparison of instantaneous stream-wise componentof velocity after t=25s for mesh M1 and M2 at Re=400.The figure on the top is for M1and lower one is for M2. The coarse mesh has run for 200 time steps and fine mesh for2500 time steps.

36

Chapter 4

Conclusions

Flow past a sphere placed in uniform stream has been analysed at low Reynolds numbers.

A stabilized finite element method is utilized to solve the incompressible Navier-Stokes

equation in primitive variables formulation. The flow features and the aerodynamic co-

eficients compare well with previous computational and experimental results. Vortex

shedding in the wake of the sphere starts at Re = 300. CD and CL show regularity in

the strength of vortices being shed at this Reynolds number. Asymmetric lift coefficient

is observed for Reynolds number in the range 300-400. Beyond Re = 400, irregularity in

the strength of vortices is observed from the plots of CD and CL. This is because hair-

pin shaped vortices are shed from only half of the sphere. Beyond Re = 700, multiple

dominant frequencies are observed in the energy spectra. Isovorticity surfaces at these

Re show non-linear interaction of vortices. Results are also compared for a coarse mesh

and a fine mesh at Re = 300 and 400. As Reynolds number is increased the difference

in the results from the two meshes are reported. Speedup is also estimated for the two

meshes. Both the meshes show sublinear speedup. Increasing the number of processors

increased the communication time between individual processors which reduced the over-

all speedup. As the number of equations is increased the speedup approaches towards

37

CHAPTER 4. CONCLUSIONS 38

being linear. Computations were also carried out for flow past a sphere with trip. With

the current mesh, oscillations are observed is presure near the surface of the sphere. A

new mesh has to be made to understand the flow physics around the cricket ball.

38

References

[1] R. Mittal and F. Najjar, “Vortex dynamics in the sphere wake,” Red, vol. 2, p. 2,

1999.

[2] H. Sakamoto and H. Haniu, “A study on vortex shedding from spheres in a uniform

flow,” ASME, Transactions, Journal of Fluids Engineering, vol. 112, pp. 386–392,

1990.

[3] S. Behara and S. Mittal, “Transition of boundary layer on a circular cylinder in

uniform flow ..”.

[4] E. Achenbach, “Vortex shedding from spheres,” Journal of Fluid Mechanics, vol. 62,

no. 02, pp. 209–221, 2006.

[5] V. Guschin, A. Kostramov, P. Matyushin, and E. Pavlyukova, “Direct numerical

simulation of the transitional separated fluid flows around a sphere,” Computational

fluid dynamics journal, vol. 10, no. 3, pp. 344–349, 2001.

[6] A. Roshko, “On the development of turbulent wakes from vortex streets,” 1953.

[7] V. Kalro and T. Tezduyar, “3D computation of unsteady flow past a sphere with a

parallel finite element method..” preprint (1997).

[8] A. Sayers and A. Hill, “Aerodynamics of a cricket ball,” Journal of Wind Engineer-

ing and Industrial Aerodynamics, vol. 79, no. 1-2, pp. 169–182, 1999.

39

REFERENCES 40

[9] R. Mehta, “Aerodynamics of sports balls,” Annual Review of Fluid Mechanics,

vol. 17, no. 1, pp. 151–189, 1985.

[10] G. Strang and G. Fix, An analysis of the finite element method. Prentice Hall, 1973.

[11] O. Ladyzhenskaya, The mathematical theory of viscous incompressible flows. Gordon

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[12] I. Babuska, “Error-bounds for finite element method,” Numerische Mathematik,

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[13] F. Brezzi, “On the existence, uniqueness and approximation of saddle-point prob-

lems arising from Lagrangian multipliers,” RAIRO, Modelisation Math. Anal. Nu-

mer., vol. 8, pp. 129–151, 1974.

[14] A. Brooks and T. Hughes, “Streamline upwind/Petrov-Galerkin formulations for

convection dominated flows with particular emphasis on the incompressible Navier-

Stokes equations,” Computer methods in applied mechanics and engineering, vol. 32,

pp. 199–259, 1982.

[15] T. Hughes, L. Franca, and M. Balestra, “A new finite element formulation for com-

putational fluid dynamics V. Circumventing the Babuska-Brezzi condition: A stable

Petrov-Galerkin formulation for the Stokes problem accommodating equal-order in-

terpolations,” Computer methods in applied mechanics and engineering, vol. 59,

pp. 85–99, 1986.

[16] T. Tezduyar, S. Mittal, S. Ray, and R. Shih, “Incompressible flow computations with

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Computer methods in applied mechanics and engineering, vol. 95, pp. 221–242, 1992.

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REFERENCES 41

[17] J. Almedeij, “Drag coefficient of flow around a sphere: matching asymptotically the

wide trend,” Powder Technology, vol. 186, no. 3, pp. 218–223, 2008.

[18] K. Watanabe, H. Kui, and I. Motosu, “Drag of a sphere in dilute polymer solutions

in high Reynolds number range,” Rheologica Acta, vol. 37, no. 4, pp. 328–335, 1998.

[19] S. Behara and S. Mittal, “Parallel finite element computation of incompressible

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41

main

Documents

past cylinder

smooth sphere

sphere wake

dierent reynolds number

high reynolds number

low reynolds numbers

lower number of elements

d mesh