Development of immersed boundary methods for isothermal and thermal flows 2

The present boundary condition-enforced IBM is established based on the fractional step technique while the body force in the modified momentum equation is implicitly determined in a way

Chapter 2

Governing Equations and Boundary Condition-Enforced Immersed Boundary Method

A new version of boundary condition-enforced IBM, which is given under the

framework of NS solver in primitive variable form, is presented in this chapter

It aims at extending the LBM solver-based IBM of Wu & Shu (2009) to the

NS solver-based IBM for an accurate evaluation of the body force The present

boundary condition-enforced IBM is established based on the fractional step

technique while the body force in the modified momentum equation is

implicitly determined in a way that the no-slip condition on the immersed

boundary is accurately satisfied The performance of the new version of IBM

is carefully examined, firstly through the classical problem of flow over a

single stationary circular cylinder, and then the flow interference between two

side-by-side circular cylinders Results from moving boundary problems such

as vortex-structure interaction around a transversely oscillating cylinder and

vortex-induced-vibration of an elastically mounted circular cylinder are also

provided as a further validation

2.1 Governing equations

Let us begin by stating the mathematical expression of the present IBM

Consider an incompressible viscous flow in a two-dimensional domain Ω

which contains an immersed object in the form of a closed curve Γ , as shown

in Fig 2.1 With the use of the IBM, the immersed object is modeled as

localized body forces acting on the surrounding fluid As a result, the IBM

formulation for the incompressible viscous flow involving immersed

objects/boundaries is expressed in the primitive variable form as

The fluid pressure p and velocity vector u are the dominating flow

variables UB is the prescribed velocity of the immersed boundary Γ ρ

and μ are the fluid density and viscosity Note that a forcing term f is

added to the right hand side (RHS) of the momentum equation (2.1) to

represent the effect of immersed object Γ The forcing term f is the

localized body force density at the fluid (Eulerian mesh) point, which is

distributed from the surface force density F at the immersed boundary

(Lagrangian) point and can be expressed as

=Γ

t s

the fluid domain and immersed boundary respectively δ(x−X( , ))s t is the

Dirac delta function responsible for the interaction between fluid and the

immersed boundary In addition, the velocity at the immersed boundary

(Lagrangian) point in IBM can be interpolated from the velocity at the fluid

(Eulerian mesh) points as

( ( , ))s t ( ) (δ ( , ))s t dV

Ω

In summary, Eqs (2.1)-(2.5) build the complete set of governing equations for

an incompressible viscous flow system involving immersed

objects/boundaries, among which Eqs (2.1)-(2.2) are the familiar

Navier-Stokes equations and Eqs (2.4)-(2.5) represent the interaction between

the fluid and the immersed objects/boundaries

2.2 Solution procedure

In IBM, the solution to Eqs (2.1)-(2.5) is frequently accomplished by making

a good use of the fractional step algorithm In a time-discrete form, the

fractional step procedure is written as:

(1) Predictor step:

Solve the normal Navier-Stokes equation for a predicted velocity field u by *

disregarding the body force terms in Eq (2.1),

In the predictor step, Eq.(2.6) is advanced to the predicted velocity field u *

under the divergence-free constraint (2.2) which couples the velocity and

pressure This constraint is the major difficulty in solving the incompressible

Navier-Stokes equations and could be successfully overcome by the popular

and well-established projection method The details on the projection method

and its implementation will be given in Section 2.3 The corrector step, as

shown in Eq (2.7), involves evaluating the unknown body force fn+1 and

updating the velocity field u to the desired one * un+1 Therefore, the

evaluation of body force poses as a crucial issue and may embody the unique

feature of the IBM The corresponding technique to determine the body force

will be illustrated in details in Section 2.4

2.3 Calculation of Predicted velocity field – Projection method

Projection method was introduced decades ago by Chorin (Chorin 1968) and

later independently by Temam (1969), as an efficient numerical device to

compute incompressible Navier-Stokes equations in primitive variable

formulation where the pressure is only present as a Lagrangian multiplier for

the incompressibility/ divergence-free constraint (2.2) Based on the Hodge

decomposition, projection method efficiently decouples the computation of

velocity and pressure in a time-splitting scheme and avoids solving the

momentum equation (2.6) and incompressibility constraint (2.2)

simultaneously Projection method proceeds in the first step to compute an

intermediate velocity field u   by using the momentum equation (2.6) and

ignoring the pressure gradient term and the incompressibility constraint (2.2)

In the second step, the intermediate velocity field u is projected onto the

space of incompressibility field to obtain the pressure and divergence-free

velocity field To be specific, its implementation is as follows:

Firstly, solve for the intermediate velocity u through Eq (2.9)

by approximating Eq (2.6) using the trapezoidal rule and dropping the

pressure gradient term The convective term, which appears in Eq (2.9), can

be approximated using the 2nd-order explicit Adams-Bashforth formula

Substituting the solution of pressure equation (2.12) into Eq (2.11) will finally

produce the predicted velocity field u*

2.4 Evaluation of Body force

The evaluation of the body force has long been the key issue for the IBM and

a number of notable strategies have been developed

2.4.1 The Conventional IBM

Early remarkable methods to calculate the body force include the well-known

penalty force scheme, feedback forcing scheme and direct forcing scheme

which are generally known as “conventional IBM” These conventional IBMs

have played an important role in the early and current development of the

IBM

2.4.1.1 Penalty force scheme

The penalty force scheme was originally proposed by Peskin (1972) to deal

with elastic boundaries on the basis of Hooke’s law and was later utilized by

Lai & Peskin (2000) to calculate the singular Lagrangian force density on

solid objects In the penalty force scheme, it is assumed that the boundary

points X of the immersed object are being attached to their equilibrium

positions X by a spring with high stiffnesse κ When the boundary moves

and deviates from its equilibrium location, a restoring force F will be

generated according to the Hook’s law

( ,t n+ )=κ( (t n+ )− e(t n+ ))

so that the boundary points will stay close to their target boundary positions

To impose the no-slip condition on the immersed boundary accurately, a large

value of stiffness κ is often required which, unfortunately, would render a

stiff system of equations and lead to a severe stability constraint However, if a

lower value of κ is utilized, the spurious elastic effects such as an excessive

deviation from the equilibrium location may arise

2.4.1.2 Feedback forcing scheme

Goldstein et al (1993) generalized the penalty force model and provided a

two-mode feedback forcing scheme

which involves a spring constant αspring and a damping constant βdamp for

the control of velocity condition at the immersed boundary This forcing term

is a reflection of the velocity difference between the desired boundary value

B

U and the interpolated u, and behaves in a feedback loop such that the

successful for low Reynolds number flows but is confronted with similar

difficulties as the penalty force model in enforcing the boundary conditions

Firstly, accurate satisfying of the boundary condition requires large values of

the spring and damping constants, which can result in numerical instability

Secondly, these two constants are flow-dependent and have to be tuned in a

semi-empirical way There is no general rule for their determination, thus

making the application of the method expensive (Fadlun et al 2000)

2.4.1.3 Direct forcing scheme

To remove the annoying empirical constants, Mohd-Yusof (1997) suggested a

forcing evaluation approach in which the body force was directly derived from

the transformed momentum equation

ρ

This method is frequently termed the direct forcing method Essentially, the

discretized momentum equation is transformed such that the forcing term is

calculated by compensating the difference between the interpolated velocities

1

( ,t n+)

u X and the desired physical velocities U XB( ,t n+1) on the boundary points In this way, the method is free from empirical parameters and no longer suffers from the numerical stability limitation, thus showing substantial improvements as compared to previous formulations Although it was initially suggested in a sharp interface method, the direct forcing scheme has been successfully generalized into Peskin’s immersed boundary method by Uhlmann (2005), who incorporated the regularized delta function into the force calculation and spreading process This strategy allows for a straightforward and smoother transfer between Eulerian and Lagrangian representations, therefore making the scheme more stable and easier to implement

However, these conventional IBMs generally compute fn+1 explicitly using the information at time level n, and flow penetration to the surface of the immersed object frequently occurs, i.e., the velocity condition on Γ is only approximately satisfied Therefore, special effort is required for an accurate evaluation of the body force

2.4.2 Boundary condition-enforced IBM

Recently, Wu & Shu (2009) proposed a novel velocity correction scheme within the framework of LBM which is proven to be effective in guaranteeing

the no-slip condition on the immersed boundary They suggested that

introducing the body force fn+1 was equivalent to making a velocity

correction which should be determined implicitly in a way that the velocity

)

),

(

(X s t

u at the boundary (Lagrangian) point interpolated from the physical

velocity u at the Eulerian points equals to the given boundary velocity UB

Therefore the basic idea of their velocity correction scheme may provide an

effective and accurate way to evaluate the body force However, their velocity

correction procedure is proposed within the framework of LBM, and it would

be worthwhile to extend it into the framework of NS solver Following the

idea in Wu & Shu (2009), the body force term fn+1 should be controlled by

+

which is derived by substituting Eq (2.7) into Eq (2.8) Note that the force

density f at the Eulerian point, as shown in Eq (2.4), is distributed from the

boundary force F through the Dirac delta function δ(x−X(s,t))

interpolation, Eq (2.18) can be reformulated to be

As a result, the correlation between Un B+1 and fn+1 is now converted to the

correlation between Un B+1 and Fn+1, and the primary concentration in the

following would become the evaluation of the boundary force Fn+1 Eq (2.19)

is in complex integral form, which can be numerically approximated by

algebraic equations as follows

Suppose that the immersed boundary is represented by a set of Lagrangian

points Xi =(X i, )Y i (i=1,2, ,M), and the fluid field is covered by a fixed

uniform Cartesian mesh xj = (x j, y j) (j=1,2, ,N) with mesh

spacing xΔ = Δ =y h Furthermore, let δ(x−X(s,t)) be smoothly

approximated by a continuous kernel distribution

1( ) (5 2 | | 7 12 | | 4 ) 1 | | 2

where Δs i is the length of the th

i boundary segment It can be observed that

Eq (2.23) form a well-defined equation system for variables Fi n+1(i=1, M)

Particularly, the equation system (2.23) for the boundary force can be written

in the following matrix form as

1

n n

n M

+ +

p + x respectively The elements of coefficient matrix [ ]AF ,

as shown in Eq.(2.25), are only related to the coordinate information of the

Lagrangian boundary points and their adjacent Eulerian points

By solving the equation system (2.24) using a direct method or iterative

method, the unknown boundary force Fi n+1(i=1, ,M) at all Lagrangian boundary points are obtained simultaneously, which are then substituted into Eqs (2.22) and (2.7) to calculate the body force 1

n j

2.5 Computational sequence

The computational sequence of the IBM solver can be summarized below Assume that the flow information at time level n is known To march the flow solution from time level n to n+1,

1) Using un as the initial flow field, solve Eqs (2.9), (2.12) and (2.11) consecutively following the projection method described in Section 2.3 to get the predicted velocity *

u ; 2) Compute the elements of matrix [ ]AF based on Eq.(2.25);

3) Solve equation system (2.24) to obtain the boundary force Fi n+1

(i= ,1 M) at all Lagrangian points and then substitute them into Eq (2.22) to get the body force n+ 1

f at Eulerian points

4) Update the predicted velocity *

u to the physical velocity n+ 1

u using Eq (2.7);

2.6 Results and Discussion

Numerical experiments conducted using the boundary condition-enforced immersed boundary solver in primitive-variable formulation are demonstrated

in this section, to provide a clear view on the performance of the present solver in solving complex flows Examples including two stationary boundary cases and two moving boundary cases are presented They are respectively the flow over an isolated stationary circular cylinder, flow interference between a pair of side-by-side circular cylinders, vortex-structure interaction around a transversely oscillating circular cylinder and vortex-induced vibration of an elastically mounted circular cylinder

2.6.1 Flow over an isolated stationary circular cylinder

Flow over an isolated stationary circular cylinder is a basic fluid dynamic problem which exhibits vastly different patterns based on the Reynolds

number

μ

ρU∞D

=

Re (U∞ is the free stream velocity, D is the cylinder

diameter) Abundant experimental (Tritton 1959; Roshko 1961; Grove & Shair 1964) and numerical results (Dennis & Chang 1970; Fornberg 1980; Braza 1986; He & Doolen 1997; Liu et al 1998; Ye et al 1999; Ding et al 2004;

Niu 2006) have been accumulated in the literature and thus it has served as a

standard case to examine the capability of new numerical methods

A schematic view of the problem is depicted in Fig 2.2, where a stationary

circular cylinder of diameter D is immersed in a uniform free-stream A

sufficiently large rectangular computational domain is used for the simulations,

whose top and bottom boundaries are equipped with slip conditions of u 0

y

∂

and v=0 The inflow boundary is specified with the prescribed free stream

velocity ( , )u v =(U∞,0), while on the outflow boundary, the natural boundary

Reynolds numbers of Re=40 and Re =100 are respectively chosen as

their representatives At each flow regime, streamlines and vorticity patterns

are first provided for a general visualization of different flow behaviors

Quantitative flow characteristics like the drag coefficient C D, lift coefficient

L

C (for unsteady case), Strouhal number (for unsteady case), recirculation

length L w/D (for steady case) behind the cylinder are then calculated and

compared with published results in the literature The drag and lift coefficients

are calculated based on the widely-used definitions

D U

D U

where F D and F L are the drag and lift forces on the cylinder surface For

unsteady flows, their time-mean values are calculated by

U∞

is a dimensionless parameter corresponding to the frequency f s with which

the vortices are shed from the body Unless otherwise specified, all the results

shown below are non-dimensionalized Additionally, the dimensionless

parameters like Reynolds number, drag and lift coefficients in all the following

examples share the same definitions as in this case

To begin the simulations, domain independence study is first conducted Three

different computational domains of sizes 24D×16D , 48D×32D and

72D×48D are examined, in which the cylinder is located at ( 8 ,8D D),

( 16 ,16D D ) and ( 24 , 24D D ), respectively A convergence criteria of

u u is set for Re=40 while the simulations for Re=100

are performed until a steady state of periodicity is achieved and lasts for at

least 20 cycles (unless otherwise specified, all other flow problems in the

thesis are simulated following similar criteria) The drag coefficients obtained using each computational domain are presented for both Re=40 and 100 in Table 2.1 (for unsteady case Re=100, it is the time-mean value) While the small domain 24D×16D produces higher drag coefficients, the results obtained using the intermediate and large domains are close to each other, indicating that the intermediate domain 48D×32D is sufficiently large to provide domain-independent results Therefore, it is chosen as our computational domain for the following simulations and discussions Subsequently, mesh independence studies are performed Five different non-uniform Eulerian meshes are tested, with resolutions of

/ 20

x y h D

Δ = Δ = = , D/ 30 , D/ 40 , D/ 50 and D/ 60 , respectively, around the cylinder Results in Table 2.2 show that the calculated drag coefficient well converges to stable values as the mesh is refined A mesh size

of h=D/ 50 is fine enough to provide a reliable solution at both flow conditions and will be used for the following simulations For the unsteady flow Re=100, time step size independence is also studied Three time step

Re= , the eddies are shed alternatively from the cylinder at Re =100

(Fig.2.4) and the famous Karman vortex street has been successfully captured

As can be clearly observed from Figs 2.3 and 2.4, there is no any penetration

of streamlines through the cylinder surface, indicating that no mass transfer happens between the fluid inside the cylinder and that outside the cylinder As

a comparison, the streamlines at Re =100 obtained using the conventional IBM (Uhlmann 2005) are provided in Fig 2.5 where streamline penetrations across the cylinder surface are quite obvious, indicating that the no-slip condition on the cylinder surface is severely violated Table 2.4 shows the drag coefficient C D and the recirculation length L w/D, together with those produced by the body-fitted methods (Dennis & Chang 1970; Shukla et al 2007) and some previous immersed boundary methods (Russell & Wang 2003; Lima E Silva et al 2003; Le et al 2008) Particularly, the results from Shukla

et al (2007) are calculated based on a very high-order scheme and the results from the abovementioned immersed boundary methods are based uniform meshes with resolutions ofh=D/ 20( Russell & Wang 2003), h=D/ 10

(Lima E Silva et al 2003) and h=D/ 18 (Le et al 2008) respectively It is apparent that both the drag coefficient and recirculation length obtained by the present method agree well with the benchmark values from the body-fitted

methods It is also observed that for this steady case, the drag coefficients derived by different immersed boundary methods are close to each other while the recirculation lengths show relatively large discrepancy

The time evolution of drag and lift coefficients for Re =100 is recorded in Fig 2.6 and shows regularly periodic oscillations, which implies the periodicity of the flow field induced by the periodic vortex shedding from the cylinder Table 2.5 provides the time-mean and fluctuating values of drag and lift coefficients and the Strouhal number St obtained in the present simulation, together with those from the literature (Li et al 1991; Liu et al 1998; Lai & Peskin 2000; Uhlmann 2005; Ji et al 2012) Once again, the results from both the body-fitted methods (Li et al 1991; Liu et al 1998) and some previously proposed immersed boundary methods which employ similar mesh resolutions (Lai & Peskin 2000; Uhlmann 2005; Ji et al 2012) are presented for comparison We can see that for the unsteady case, the drag coefficient produced by the immersed boundary method is always a bit higher than the body-fitted method due to the usage of the discrete delta function However, it is obvious that our result is closer to the reference ones (Li et al., 1991; Liu et al., 1998) The lift coefficient given by the present method, as can

be seen, also matches quite well with the benchmark values provided While the Strouhal numbers produced by some previous immersed boundary methods are 1%-4% higher than those from the body-fitted methods, our result

achieves a good agreement

2.6.2 Flow interference between a pair of side-by-side circular cylinders

Flow around a pair of side-by-side stationary circular cylinders has been found

to exhibit abundant flow patterns (Chang & Song 1990; Ravoux et al 2003; Kang 2003; Mizushima & Ino 2008) The vortices shed from the two cylinders interact dynamically, generating much more complicated wake behaviors than those behind an isolated cylinder Therefore this topic is a favorable example

to further test the performance of our new method Generally, the flow is governed by two dimensionless parameters: the Reynolds number Re and gap ratio G D/ , with the latter one much more significant and sensitive Here

G denotes the gap distance between the cylinder centers Depending on various geometrical configurations of the two cylinders, the flow characteristics can be quite different from each other Our simulations here choose several representative gap ratios and focus on a fixed Reynolds number

of Re 100= The schematic diagram is plotted in Fig 2.7, where the computational domain has a size of 50D×40D The two side-by-side cylinders are located at (15D , 20D G− / 2 ) and (15D , 20D G+ / 2 ), respectively In our simulations, the whole domain is discretized by a non-uniform mesh with a fine resolution of Δ = Δ = =x y h D/ 50 inside a local region around the cylinders Meanwhile, a time step size of Δ =t 0.001

is used for the time intergration The flow details in the near wake regions in

forms of streamlines and vorticity contours as well as the lift and drag coefficients are presented and quantitatively compared with the well-established results in the literature

Fig 2.8 plots the instantaneous streamlines and vorticity contours in the near-wake region behind the cylinder-pair at G D/ =3 Consistent with the observations in Chang & Song (1990) and Ding et al (2007), the two vortex streets developeing behind the cylinder pair are apparently symmetric, falling

in an anti-phase synchronization flow regime indicated by Kang (2003) The symmetric feature also manifests itself from the time histories of drag and lift coefficients presented in Fig 2.9: while the drag coefficients of the upper and lower cylinders are almost in the same phase, the lift coefficients of the two cylinders have a 180° phase difference The time-mean lift and drag coefficients together with the Strouhal number are well calculated and listed in Table 2.6 Their comparison with the reported results of Chang & Song (1990) derived using the body-fitted method and those of Ding et al (2007) using the mesh-free method exhibits good agreement

With a systematic change of the gap ratio G D/ , different flow patterns are observed Simulations are also carried out at four other gap ratios of

G D = , 1.7, 2.5 and 4, as presented in the following Instantaneous streamlines and vorticity contours shown in Fig 2.10 reveal four completely