numerical simulation of fluid flow over a … · flow over a forward- backward facing step using...

NUMERICAL SIMULATION OF FLUID FLOW OVER A FORWARD-

BACKWARD FACING STEP USING IMMERSED BOUNDARY METHOD

C.A. SALEEL1, A. SHAIJA2 AND S. JAYARAJ 3

Mechanical Engineering Department, National Institute of Technology, Calicut,

Kerala-673601, India [email protected] 1

[email protected] 2

[email protected] 3 http://www.nitc.ac.in

Abstract: This paper details immersed boundary method (IBM) to facilitate numerical investigation of the laminar forced fluid flow over a forward-backward facing step through a 2D channel by treating step as an immersed boundary (IB). The present numerical method is based on a finite volume approach on a staggered grid together with a fractional step method. When an IB (step) is present, both momentum forcing and mass source terms are applied on the body surface to satisfy the no-slip boundary condition on the immersed boundary and to satisfy the continuity for the mesh containing the immersed boundary respectively. In the IBM, the necessity of an accurate interpolation scheme satisfying the no-slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary. Results are presented with respect to Reynolds numbers for different sizes of the step.

Keywords: Immersed Boundary Method, Forward-Backward Facing Step, Momentum Forcing, Mass Source/Sink.

1. Introduction

Numerical simulations are becoming a most important drive for the design and analysis of complex systems. The advancement of computing urges engineers to include high fidelity computational fluid dynamics (CFD) in the design and testing tools of new technological products and processes. Numerical simulations are now recognized to be a part of the computer-aided engineering (CAE) spectrum of tools used extensively today in all industries, and its approach to modeling fluid flow phenomena allows equipment designers and technical analysts to have the power of a virtual wind tunnel on their desktop computer. Computational simulation software has evolved far beyond what Navier`, Stokes or Da Vinci could ever have imagined. It has become an indispensable part of the aerodynamic and hydrodynamic design process for planes, trains, automobiles, rockets, ships, submarines, micro-electromechanical systems (MEMS), Lab-on-Chip (LOC) devices and so on; and indeed any moving craft or manufacturing process that has been devised so far. The advantage of numerical simulation over experimentation is well understood. Despite with these developments some primary issues like accuracy, computational efficiency and ability to handle complex geometries of fluid dynamic simulations are still predominant. A brief and concise literature survey on the immersed boundary method, the flow between parallel plates flows and the forward-backward facing step flows are furnished here. The present numerical method is validated by investigating flow between parallel plates and found that numerical results at the fully developed region are well in agreement with the analytical solution. The governing equations and their numerical solution methods, including the boundary conditions employed are briefed in the next Section. The results and discussion are provided separately thereafter.

1.1. Immersed Boundary Method (IBM)

Fluids flows over complex geometries are very common in engineering problems, and the major difficulty arise in how to represent the body, its moving walls and its interaction with the fluid. The most accustomed approach

C.A. Saleel et al. / International Journal of Engineering Science and Technology (IJEST)

ISSN : 0975-5462 Vol. 3 No.10 October 2011 7714

is using Neumann and Dirichlet boundary conditions to represent the body geometry. Therefore, if the geometry is complex, treatment will have a hard and probably a complicated work. This difficulty grows up if the body has a poignant and deformable geometry. In short, treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations. Both the unstructured grid method and IBM are used for simulating flow with complex geometries.

In addition to the above mentioned two methods, some authors have proposed different methods to treat this kind of problem. For example, Harlow and Welch (1965) proposed the marker and cell (MAC) approach. In this method the fluid region on one side of the boundary is identified by markers, while on the other side of the boundary, which can be fluid or solid, is identified by another marker. It requires huge storage space and CPU time.

The phrase “immersed boundary method” was first appeared in literature in reference to a method developed by Peskin (1972). A force term added to the Navier-Stokes equation is in charge to promote the interaction between fluid-solid interactions. Varieties of innovative ideas have been proposed to calculate this force term and each initiative leads to a particular immersed boundary technique. Originally this technique was used to simulate cardiac mechanics and associated blood flow. The legendary feature of this method was that, the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart and a novel procedure was imitated for imposing the effect of the immersed boundary (IB) on the flow. The boundary conditions are imposed not in a straight forward manner in IBM. Since Peskin’s introduction about this method, there arise numerous modifications and refinements to this technique. Now a number of variants of this approach exist. The major advantages of the Immersed Boundary Method include computer memory and CPU time savings. Also easy grid generation is possible with IBM compared to the unstructured grid method. Even moving boundary problems can be handled using IBM without regenerating grids in time, unlike the structured grid method. It is to be noted that generation of body conformal grids (structured or unstructured) is habitually very cumbersome.

The key factor in developing an IB algorithm is the way of imposition of boundary conditions on the IB and the same distinguishes one IB method from another. In the former approach, which is termed as “continuous forcing approach”, the forcing function is incorporated into the continuous equations before discretization, where as in the latter approach, which can be termed the “discrete forcing approach”, the forcing function is introduced after the equations are discretized. An attractive feature of the continuous forcing approach is that it is formulated independent of the underlying spatial discretization. On the other hand, the discrete forcing approach very much depends on the discretization method. However, this allows direct control over the numerical accuracy, stability, and discrete conservation properties of the solver [Mittal and Iaccarino (2005)]. The merits of continuous forcing approach are its attractiveness for problems with elastic boundaries, closeness to the physics of the problem; hence relatively easy to conjure up the realistic flow problems especially high feasibility for successful simulation of biological and multiphase flows. The demerits of the aforementioned method include development of “stiff” numerical systems due to the presence of rigid Immersed Boundary in flow problems. Here satisfactory results have only been attained for low Reynolds number flows with moderate unsteadiness. Smoothing of the forcing function prohibits the sharp representation of the IB, which is not acceptable at high Reynolds numbers. The method also necessitates the computation in substantial amount of grid points located inside the body which simply results in unnecessary extra computation time. The merits of discrete forcing approach include

(i) Suitability for flows around rigid bodies, (ii) Handling of higher Reynolds number flows, (iii) Absence of stiffness or user defined parameters that can impact the stability of the method, (iv) The ability to represent sharp IB by imposing the boundary conditions directly on the numerical

scheme and (v) Computation of flow variables inside a rigid body becomes unnecessary.

Where as its demerits are need for a pressure boundary condition on the IB and moving boundaries are harder to deal with than in continuous forcing IBMs. A review about Immersed Boundary Methods (IBM) encompassing all variants is cited by Mittal and Iaccarino (2005). Feedback forcing method is applied to represent a solid body by Goldstein et al. (1993) which induced spurious oscillations and restricted the computational time step associated with numerical stability.



Yusof (1997) proposed a different approach to evaluate the momentum forcing function in a spectral method, and his method does not require a smaller computational time step, which is an important benefit of this method over preceding methods. The Discrete Immersed Boundary Finite Volume Method [Kim et al. (2001)] used to simulate the present problem (i.e., to simulate the channel flows with obstructions) is based on a finite volume approach on a staggered mesh together with a fractional step method. The obstruction is treated as an Immersed Boundary. Both momentum forcing and mass source are applied on the body surface or inside the body to suit the no-slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary. In the IBM, the choice of an accurate interpolation scheme satisfying the no-slip condition on the IB is important

1.2. Flow between parallel plates

In recent times, escalating attention has been focused on the parallel plate ducts since it is encountered in many of the energy related applications. Significant application of this configuration includes flat plate solar energy collection systems, cooling of electronic systems, MHD energy conversion systems, MEMS etc. Hence computational simulation of the same is widely focused topic among the CFD community. Electronic components are mounted on PCBs like an array, forming vertical flat channels through which coolants are forced to flow. The details of the same are seen in Otani and Tanaka (1975) and also in EPP staff report (1981). The coolant may be re-circulated by forced convection for large applications. The majority of compact heat exchangers exhibit laminar flow because of their low hydraulic diameters. The effects of the entrance region cannot be neglected as the flow channel length is short. Comprehending the flow development is crucial in the analysis of the flow of heat. The research related to flow and heat transfer through parallel plate channels has been well cited by Peterson and Ortega (1990) in their investigation of thermal control of electronic equipment. The more up to date work reported by du Toit (2000), Spiga and Morini (1996), Schwiebert and Leong (1995), Hao and Chung (1995) and Inagaki and Komori (1995) exposed the need for computational simulation of flow and heat transfer through channels between parallel plates. This type of numerical simulation would help researchers and engineers, who are working in similar or related applications, for the analysis and study of flow and heat transfer in parallel plate channels. Also, introduction of immersed solid objects between the plates in the flow path shall enhance the heat transfer.

1.3. Forward-Backward Facing Step Flows

The study of flow separation is an important issue in fluid dynamics, which results in a total change of the flow field. The flow separation and recirculation caused by a sudden contraction in a channel, in the form of a forward-backward facing step is an example for the same. It has got not only a lot of engineering applications, for example for the design of heat transfer devices, such as cooling systems for electrical equipment or high performance heat exchangers, but is also observed in nature, for example in rivers, in lakes and flows due to erosion over grooves, which were formed by previous glaciations. The existence of flow separation and subsequent reattachment caused by a sudden expansion or compression in flow geometry, such as forward- backward facing steps greatly influences the mechanism of heat transfer. It is well established that there is a large variation of the local heat transfer rate within separated flow regions, and substantial heat transfer enhancement may result in the reattachment zone. Thus, in thermal engineering applications where cooling or heating is required, it is essential to understand the basic mechanism of heat transfer in such flows. These heat transfer applications appear in cooling systems for electronic equipment, combustion chambers, cooling of nuclear reactors, chemical processes and energy systems equipment, high-performance heat exchangers, and cooling passages in turbine blades. A great deal of mixing of high- and low-energy fluid occurs in the reattached flow region in these heat transfer devices, thus affecting their heat transfer performance. For the case of the forward- backward facing step, depending on the ratio of the boundary-layer thickness at the step to the step height, one or two or three recirculating regions may develop (one upstream, the other downstream from the step and the third one is immediately above the step) . Also the flow separates at the right hand upper sharp corner of the step, causing a recirculating region to develop behind the step. Addad et al.(2003) studied the feasibility of using a commercial CFD code for large eddy simulation (LES) of a forward–backward facing step at Reynolds number based on the channel height, Reh = 1.7x105 for



acoustic source identification. Leclercq et al. (2001) investigated aerodynamic flow over forward-backward facing step and found that three recirculation eddies are developed for higher Reynolds numbers.

Abu-Mulaweh H. I. (2009) made an experimental study about the effects of forward-backward facing steps on a turbulent buoyancy-dominated mixed-convection flow over a flat plate. His results reveal that the introduction of forward-backward facing steps increases the turbulence intensity of the velocity and temperature fluctuations downstream of the step. Eventhough the backward-facing step and forward-facing step has been extensively studied for the flow field, the flow over forward-backward facing step problem is relatively new in the study of flow field and even in acoustics. This makes the present study more striking, particularly the use of Immersed Boundary Method to obtain the numerical solution

2. Solution Methodology

2.1. Immersed Boundary Method

To explain the concept of immersed boundary method, consider the simulation of flow past a solid body shown in Figure 2 [Mittal and Iaccarino (2005)].

Fig. 1 Schematic showing a generic body past which flow is to be simulated.

The body occupies the volume Ωb with boundary Γb. The body has a characteristic length scale L, and a

boundary layer of thickness δ develops over the body. In conventional approach, this would employ structured or unstructured grids that conform to the body. Generating these grids proceeds in two sequential steps. First, a surface grid covering the boundaries Γb is generated. This is then used as a boundary condition to generate grids in the volume Ωf occupied by the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the governing equations is transformed to a curvilinear coordinate system aligned with the grid lines Ferziger and Peric [1996]. Because the grid conforms to the surface of the body, the transformed equations can then be discretized in the computational domain with relative ease. If a finite-volume technique is employed, then the integral form of the governing equations is discretized and the geometrical information regarding the grid is incorporated directly into the discretization. If an unstructured grid is employed, then either a finite-volume or a finite-element methodology can be used. Both approaches incorporate the local cell geometry into the discretization and do not resort to grid transformations.

Now consider employing a non-body conformal Cartesian grid for this simulation, as shown in Figure 3 [Mittal and Iaccarino (2005)]. In this approach the immersed boundary (IB) would still be represented through some means such as a surface grid, but the Cartesian volume grid would be generated with no regard to this surface grid. Thus, the solid boundary would cut through this Cartesian volume grid. Because the grid does not conform to the solid boundary, incorporating the boundary conditions would require modifying the equations in the vicinity of the boundary. Precisely what these modifications are is the subject matter of IBM. However, assuming that such a procedure is available, the governing equations would then be discretized using a finite-difference, finite-volume, or a finite-element technique without resorting to coordinate transformation or complex discretization operators.



Fig. 2 Schematic of body immersed in a Cartesian grid on which the governing equations are discretized.

2.2. Governing Equations

The general governing equations for unsteady, incompressible, viscous flow between parallel plates are written as

2

2

( ) 1(1)

Re

0

i ji i

i

j i j j

i

i

u uu upf

t x x x x

uq

x

where ix are the Cartesian coordinates, iu are the corresponding velocity components, p is the pressure,

if are the momentum forcing components defined at the cell faces on the immersed boundary or inside the body, and q is the mass source/sink defined at the cell center on the immersed boundary or inside the body. All the variables are non-dimensionalized by the bulk average velocity of the inlet flow, Ub and the length scales are non-dimensionalised by the channel height at the downstream, H. The only dimensionless number appearing in the governing equations is the Reynolds number. For the flow problem considered, the following definition is used for the Reynolds number, Re.

Re (3)bU H

Here and are the density and the dynamic viscosity, respectively

2.3. Size of Flow Domain and Boundary Conditions

Fig.3. Sketch of the flow configuration and definition of length scales for flow between parallel plates

ZX

Y

1, 0,

0

u v

p

x

0,

0

Outlet

u

xv

0, 0,

0

Bottom wall

u v

p

y

0, 0,

0

Top wall

u v

p

y

H



Figure 3 depicts the geometry of the flow domain for flow between parallel plates with finite distance in between the channel, which is small compared to its length and width. Hence the flow through this channel is assumed to be two dimensional. Figure 4 depicts the two-dimensional channel with a forward-backward facing step with finite distance in between the channel, which is small compared to its length and width. Hence the flow through this channel is assumed to be two dimensional. The geometry consists of an upstream wall, a forwardfacing step, a backward-facing step, and a downstream wall. In both the flow domains, the flow is assumed as unsteady and laminar and buoyant forces are negligible compared with viscous and pressure forces. The following boundary conditions are assumed for the computational simulation. Inlet: In order to simulate a fully developed laminar channel flow, a uniform velocity profile with is prescribed at the channel inlet for the present model. Cross stream velocity is equal to zero. The Neumann boundary condition is assumed for pressure. Outlet: Fully developed velocity profile is assumed at the outlet. Pressure boundary condition is not specified. Walls: No slip condition (u=0 and v=0) for velocity and Neumann boundary condition for pressure.

Fig.4. Sketch of the flow configuration and definition of length scales for forward-backward facing step.

The time-integration method used to solve the above equations is based on a factional step method

where a pseudo-pressure is used to correct the velocity field so that the continuity equation is satisfied at each computational time step. In this study, a second-order semi-implicit time advancement scheme (a third order Runge-Kutta method for the convection terms and a second order Crank-Nicholson method for the diffusion terms) is used. Eq. (1) and (2) are expanded as follows:

1 11 1 2

2

ˆˆ( ) ( ) 2 ( ) ( ) (4)

ˆ1( ) (5)

2

k k kk k k k ki i

k i k i k k i k i ii

kkki

i i k i

u u pL u L u N u N u f

t x

uq

x x t x

The velocity and pressure values are calculated using the following equations

21

ˆ 2 (6)

(7)Re

kk k

i i ki

kk k k k

j j

u u tx

tp p

x x

L(ui) and N(ui) appearing in Eq. (5) corresponds to 21

( ) (8)Re

( ) (9)

ii

j j

i ji

j

uL u

x x

u uN u

x

Where ˆiu is the intermediate velocity, is the pseudo-pressure, t is the computational time step, k

the sub-step index, and k , k and k are the coefficients of RK3 (Third order Runge-Kutta) whose values are

H

Z

0,

0

Outlet

u

xv

0, 0,

0

Bottom wall

u v

p

y

0, 0,

0

Top wall

u v

p

y

X

Y

1, 0,

0

u v

p

x



1 1 1

2 2 2

3 3 3

4 8, , 0 (10)

15 15

1 5 17, , (11)

15 12 60

1 3 5, , (12)

6 4 12

A discrete-time momentum forcing function, if should be appended with the Navier-Stokes equations to

treat the immersed boundary (IB) as a kind of forcing so that it mimic the effect of IB. Here it is extracted from the literature presented Yusof (1997). This forcing function is incorporated to satisfy the no-slip condition on the immersed boundary (IB) and is applied only on the immersed boundary or inside the body. In the absence of IB,

if should be made equal to zero. The location of points, where the forcing function has to be introduced, is determined in a similar fashion as that of the velocity components defined on a staggered grid. And a mass source/sink term q is introduced for the cell containing the immersed boundary to satisfy the mass conservation. The mass source/sink term is applied to the cell center on the immersed boundary or inside the body. The way of finding the momentum forcing function, if and the mass source/sink term q is explained as follows.

2.4. Momentum Forcing and Interpolation for the Velocity

To obtain ˆiu , from Eq. (5), the momentum forcing kif must be determined in advance such that ˆiu satisfies the

no-slip condition on the immersed boundary. When Eq. (1) is provisionally discretized explicitly in time (RK3 for the convection terms and forward Euler method for the diffusion terms) to derive the momentum forcing value, we have

1 11 1 22 ( ) 2 ( ) ( ) (13)

K K KK K K Ki i

i i i iK K K Ki

u u pL u N u N u ft x

Rearranging Eq. (14) results in the following equation for kif at a forcing point,

1 11 1 22 ( ) 2 ( ) ( ) (14)

K K KK K K Ki i

i i i iK K Kki

U u pf L u N u N ut x

Here k

iU is the velocity to be obtained at a forcing point by applying momentum forcing. In the following,

kiu ( ˆk

iu in Eq. (5)) indicates the velocity at a grid point nearby the forcing point updated from Eq. (14) with

0kif to determine K

iU using the linear interpolation.

In case of the no-slip wall, kiU is zero as and when the forcing point coincides with the immersed boundary.

However, in general the forcing point exists not on the immersed boundary but inside the body, and thus an interpolation procedure for the velocity k

iU is required. In the present study, second-order linear interpolations are used, and Figure 6 shows the schematic diagrams for the calculation of interpolation velocity when the backward facing step is considered as the IB.

KBu

KCu

KAu

1KU

Ay

By

h

hKAv

KCv

1KV

1h

1Ay

1Kv

1Ku 2

Ku

2Kv

x

y

Fig. 5 Stencil for the linear interpolation scheme in the vicinity of backward facing step (IB) which shows instantaneous velocity,

interpolation velocity, forcing points, etc.



To explain the interpolation scheme distinctively with respect to Fig.6, the following second-order linear interpolation is considered

1 (15)K KCU u

For 0 Ah y , KCu is obtained from a linear interpolation between K

Au and the no-slip condition at the IB,

whereas for A By h y , KCu is obtained from K

Au and KBu . That is

1

1

0 16

( ) ( )17

K KA A

A

K KK B A A B

A BB A

hU u for h y

y

y h u h y uU for y h y

y y

The exact position of IB with respect to grids is determined and either Eq. (17) or Eq. (18) has to be used to calculate the U-interpolation velocity. The same scheme is applied to the y-component velocity. In the case of v-velocity, variant of Eq. (17) and Eq.(18) may be written as follows:

11 1 1

1

1 1 1 11 1 1 1

1 1

0 18

( ) ( )19

K KA A

A

K KK B A A B

A BB A

hV v for h y

y

y h v h y vV for y h y

y y

The exact position of IB with respect to grids is determined and either Eq. (19) or Eq. (20) has to be used to calculate the V-interpolation velocity.

2.5. Mass Source and Continuity Equation

The procedure of obtaining the mass source Kijq in Eq. (6) is explained in this section. Consider the star marked

two-dimensional cell shown in Fig. 6, where 1Kv is the velocity components inside the body and 1

Ku , 2Ku and

2Kv are those outside the body. For the rectangular cell containing only fluid, the continuity reads

1 2 2 0 (20)K K Ku y v x u y

Meanwhile, for the rectangular cell containing both the body and the fluid the continuity equation becomes

1 1 2 2 (21)K K K K Ku y v x q x y u y v x

From Eq. (21) and Eq. (22), the mass source K

ijq is obtained as 2

K

K vq

y

. In general,

(22)KijK

ij

vq

y

Note that 2Kv is unknown until equations (6) and (7) are solved and thus we use 2ˆ

Kv instead of 2Kv and is updated

as and when 2Kv is being found out. Therefore, in general the mass source K

ijq is defined as ˆ

(23)KijK

ij

vq

y

2.6. Solution Procedure

For the spatial discretization of Eq. (5) to (8) a finite volume approach on a staggered grid together with a fractional step method was employed. Being a CFD method, the Finite Volume Method (FVM) describes mass, momentum and energy conservation for solution of the set of differential equations considered. The FVM is comparable to other approximated numerical methods such as Finite Difference Method (FDM) and Finite Element Method (FEM). It is characterized by the partition of the spatial domain in a finite number of elementary volumes for which are applied the balances of mass, momentum and energy. The approximated equations for the method can be obtained by two approaches. The first consists in applying balances for the elementary volumes (finite volumes), and the second consists in the integration spatial-temporal of the conservation equations. In this work, the latter approach is followed.

The convection and diffusion terms were evaluated using a central differencing scheme of second-order accuracy. Solution of non-dimensional u and v are made possible in Alternating Direction Implicit (ADI)



approximate factorization method clubbed with powerful and accurate Tri-Diagonal Matrix Algorithm (TDMA). The pressure solver is based on Successive over Relaxation (SOR) method. The numerical code is developed using Digital Visual FORTRAN (DVF) and a detailed flow chart is shown in Figure 7, based on which the computer code is developed. To generate all results, the respective computer code is executed with an accuracy of 10-6.

Fig. 6 Flow chart for the Immersed Boundary Method

3. Results and Discussions

3.1. Numerical Simulation of Flow Between Parallel Plates

Firstly, the present numerical method is validated by investigating the flow bwteen parallel plates and comparing the channel outlet streamwise velocity profile is compared with the theoretical solution of the same. The results are generated for flow between parallel plates at Reynolds numbers 1, 2, 10, 20, 50,100. To have a theoretical solution at the channel outlet (i.e., for the hydro-dynamically fully developed flow),

0 (24)U

X

Hence, the axial (x-direction) momentum equation can be written as

START

Define Grid size, Define RK3 coefficients, Assign initial values to velocity, pressure and pseudo-pressure, Set iteration no=0.0, Set BCs

Iteration Number = Iteration Number+1

Determine mass source term (IB), Solve pseudo-

pressure using SOR

k (fractional step index) loop = 1 to 3 Determine momentum forcing (IB), Solve intermediate u-velocity, Solve intermediate v-velocity and Update intermediate velocity BCs

Converge?

Update the pseudo-pressure BCs, Determine the

final u, v, p with the converged pseudo-pressure

Converge?

END

If k>3

Yes

Yes

No

No

Yes

No



2

2(25)fd

fd

U P

XY

The left-hand side of the above equation is function of ‘Y’ only while its right-hand side is function of X only. This implies that both sides should be a constant. Integrating the above equation and applying the non-slip conditions at the wall (Y=0 and Y=1), one can obtain an expression for the fully developed velocity profile as

21( ) (26)

2fdfd

dPU Y Y Y

dX

Substituting for ‘U’ from above equation into the following integral form of the continuity equation,1

0

1UdY ,

one can evaluate the value of the fully developed pressure gradient. This fully developed pressure

gradientfd

dP

dX

, was found to be (-12). So, the analytical fully developed velocity profile can be given as

2( ) 6 (27)fdU Y Y Y

The numerically obtained developing axial velocity profile should approach asymptotically this analytical fully developed axial velocity profile. So, this analytical profile is used as a check for the validity of the numerical code developed. Figure 7 shows the excellent agreement between the numerical and analytical streamwise velocity. The numerical streamwise velocity profile at the exit of the channel is independent of Reynolds numbers as shown in Eq. 27.

Fig.7. Excellent agreement between the analytical and numerical streamwise velocity profiles at the exit of the channell Figure 8 shows the streamwise velocity contour for the flow between parallel plates for the above mentioned Reynolds numbers. It is clearly seen that the hydrodynamic emtry length is gradually increasing with increase of Reynolds numbers. Figure 9 depicts the variation of non-dimensional hydrodynamic entrance length with respect to Reynolds numbers. The variation is almost in an exponential manner. Streamwise velocity contour reveals the hydro dynamically developing region at the entrance of the channel and the hydro dynamically developed region there after. Inspecting Figures 8, one can conclude that at the early stages downstream the entrance, the fluid adjacent to the walls decelerates due to the formation of the two hydrodynamic boundary layers. Consequently, as a result of continuity principle, fluid outside these two boundary-layers accelerates. Due to this action, at the entrance region, a transverse velocity component is having an influencing value that sends the fluid away from the two plates outside the two boundary-layers and towards the centre line between the two walls. The development of the transverse velocity component show how big this velocity component is at the early stage in the entrance region due to the formation of the two hydrodynamic-boundary-layers. However, this action gradually decays with further increase in the axial distance downstream the entrance and finally the value of transverse velocity vanishes to near zero when the flow becomes hydro-dynamically fully developed.



Fig. 8 Streamwise velocity contours for the flow between parallel polates

Fig. 9 Variation of hydrodynamic entry length with respect to Re values

Energy Engineers are not only frequently concerned with the details of the fluid velocity but also with the pressure drop. Pressure variations show the development of the pressure with non-dimensional channel length (X) from the entrance till the fully developed conditions are achieved. When the flow becomes hydro-dynamically fully developed (far away downstream the entrance) the pressure drop is due to the viscous shear action only. At this stage the pressure drop can be obtained as explained. Under the assumption of the whole channel fully developed flow right from the entrance, the linear variation of fdP with X can be calculated using

the analytically obtained value offd

dP

dX

. For a developing flow in the entry region, the pressure drop is due to

both inertia and viscous actions. Consequently, the dimensionless pressure is a nonlinear function of X. However, pressure P should finally (at large values of X) follow a linear relationship with X having the

analytically obtained slope, .fd

dP

dX

3.2. Computational Simulation ofFlow Over Forward-Backward Facing Step Secondly, the flow over a forward-backward facing step is numerically simulated using the present method. This configuration has prospective applications and appears frequently in engineering situations, particularly with heat transfer applications. Also, it has applications in hydraulics, oceanography and dam analysis. The obstructions are used as turbulence enhancers in order to generate the increased heat transfer coefficients and also as eddy generators. Moreover such situation exists also in the oxygenator which is a medical device, which is capable of exchanging oxygen and carbon dioxide in the blood of human patient in surgical procedures.

Re=1 Re=2 Re=10

Re=20 Re=50 Re=100



Rectangular obstacles excite secondary flows in micro-channels thereby greatly enhance the mixing of two liquids flowing in a layered fashion through these micro-channels. In addition to Reynolds numbers, the most dominant parameters that manipulate the flow phenomena are the shapes and sizes of obstruction geometry and the gap between the obstruction and the top wall of the test channel. Authority of these flow parameters is investigated using velocity contours and stream lines plotted for various cases. The velocity contours and the stream line plots provide the quantitative and qualitative information regarding the flow features. Keeping this in mind, three geometrically different step configurations are considered arbitrarily. They are height and length of the step, which are normalized with channel height. 3.2.1. Forward-backward facing step (height =0.25) Flow over forward-backward facing step of non-dimensional height = 0.25 is investigated first numerically for the Reynolds number value of 1, 10, and 50. Figure 10 describes the stream wise velocity contours of the flow over forward-backward facing step through the channel in the same order of the aforesaid Reynolds number values. The contour unveils that maximum stream wise velocity is exactly above the step. As the numerical experimentation is on the way two recirculation eddies are seen for some cases and to differentiate one from another, recirculation eddies nearer to the obstruction is named as primary recirculation eddies and the recirculation eddies nearer to the top wall is named as secondary recirculation eddies. For the present configuration, the primary recirculation eddies are negligibly absent when Re=1 and Re=10 and are predominant in the down stream side of the step when Re=10 and 50. Size of the recirculation eddies are linearly increasing with the value of the Reynolds number.

Fig.10. streamwise velocity contours for the flow over a forward-backward facing step (height=0.25) at Re=1, 10, and 50. Figure 11 shows the streamline plots for the same geometry as in same ordinal status of Reynolds numbers. It is found that primary recirculation eddies being formed for Re=1,10 and 50. The size of the recirculation eddies or vortex in the downstream side is larger for Re=50. It can be seen from the numerically predicted velocity contours and stream lines for different Reynolds numbers that the shear layers separating from the edges of the step reattached on the bottom wall of the test channel at different positions. The reattachment length increases linearly with Reynolds numbers. The values of reattachment length are more clearly noticed from the stream line plots (as shown in Figure 12). It can also be seen that a low pressure zone is formed at the vicinity immediately after the step in the channel which triggers the flow to generate recirculation eddies at Re = 10 and 50. Continuity is satisfied by adjusting the values of transverse velocity wherever required.

Fig.11. streamlines in the vicinity of forward-backward facing step (height=0.25) at Re=1, 10, and 50.

Non-Dimensional Channel Length

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3.2.2. Forward-backward facing step(height =0.5)

Flow over forward-backward facing step of non-dimensional height = 0.5 is experimented numerically next for the Reynolds number value of 1, 10, and 50. Figure 12 describes the stream wise velocity contours of the flow over forward-backward facing step through the channel in the same order of the aforesaid Reynolds number values. The contour unveils that maximum stream wise velocity is exactly above the step. As the numerical experimentation is on the way two recirculation eddies are seen for some cases and to differentiate one from another, recirculation eddies nearer to the obstruction is named as primary recirculation eddies and the recirculation eddies nearer to the top wall is named as secondary recirculation eddies. For the present configuration, the primary recirculation eddies are negligibly absent when Re=1 and Re=10 and are predominant in the down stream side of the step when Re=10 and 50. Size of the recirculation eddies are linearly increasing with the value of the Reynolds number. Figure 13 shows the transverse velocity contours for the same geometry and Reynolds numbers. Apart from the inlet, the maximum and minimum value of transverse wise velocity is located immediately before and after the tips of the step. The minimum value of the transverse velocity immediately after the tip physically demands that the flow needs some stream wise distance to adjust and become fully developed. As the Reynolds number increases, there occurs an increase in area of realm where maximum and minimum cross-stream velocity demanding more stream wise distance downstream of the step to get fully developed.

Fig.12. streamwise velocity contours for the flow over a forward-backward facing step (height=0.5) at Re=1, 10, and 50.

Figure 14 shows the streamline plots obtained from the flow field for the same geometry as in same ordinal status of Reynolds numbers. It is found that two recirculation eddies being formed for Re=1 and 10. For Re = 50,three recirculation eddies are developed around the step. The recirculation eddies developed immediately above the step reveals that the results are in good agreement with that of Leclercq et al. (2001).The size of the recirculation eddies or vortex in the downstream side is larger for Re=50. It can be seen from the numerically predicted velocity contours and stream lines for different Reynolds numbers that the shear layers separating from the edges of the step reattached on the bottom wall of the test channel at different positions. The reattachment length increases linearly with Reynolds numbers. The values of reattachment length are more clearly noticed from the stream line plots (as shown in Figure 15). It can also be seen that a low pressure zone is formed at the vicinity immediately after the step in the channel which triggers the flow to generate recirculation eddies at Re = 10 and 50. Continuity is satisfied by adjusting the values of transverse velocity wherever required.

Fig.13. Transverse velocity contours for the flow over a forward-backward facing step (height=0.5) at Re=1, 10, and 50.

Re=1 Re=50 Re=10

Re=1 Re=1 Re=50



Fig.14. streamlines in the vicinity of forward-backward facing step (height=0.5) at Re=1, 10, and 50.

3.2.3. Forward-backward facing step(height =0.75)

Finally, the flow over forward-backward facing step of non-dimensional height = 0.75 is studied numerically for the Reynolds number value of 1 and 10. Figure 15 describes the stream wise velocity contours of the flow over forward-backward facing step through the channel in the same order of the aforesaid Reynolds number values. The contour unveils that maximum stream wise velocity is exactly above the step. As the numerical experimentation is on the way two recirculation eddies are negligibly small for both the Reynolds numbers. Figure 15 also shows the streamline plots for the same geometry as in same ordinal status of Reynolds numbers. It is found that primary recirculation eddies being formed for Re=1 and almost nil for Re = 10. The size of the recirculation eddies or vortex in the upstream and downstream side is equal for Re = 10. It can be seen from the numerically predicted velocity contours and stream lines for different Reynolds numbers that the shear layers separating from the edges of the step glides through the step and reattached at the base of the step on the bottom wall of the test channel. It can also be seen that a low pressure zone is formed at the vicinity immediately before and after the step in the channel which triggers the flow to generate recirculation eddies at Re = 1. As before, it is seen that continuity is getting satisfied by adjusting the values of transverse velocity wherever required.

Fig.15. Streamwise velocity and streamlines for flow over forward-backward facing step (height=0.75) at Re=1 and 10.


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4. Conclusion

Immersed boundary method is adopted to validate a relevant fluid mechanics benchmark problem, the flow between parallel plates and is extended to conduct numerical experimentation of flow over forward-backward facing step in channels. The present method is based on a finite volume approach on a staggered mesh together with a fractional-step method. The momentum forcing and the mass source/sink are applied on the body surface or inside the body to satisfy the no-slip boundary condition on the immersed boundary and the continuity for the cell containing the immersed boundary, respectively. A linear interpolation scheme is used to satisfy the no-slip velocity on the immersed boundary, which is numerically stable regardless of the relative position between the grid and the immersed boundary. The present algorithm is ideally suited to low Reynolds number flows as well. Predictions from the numerical model have been compared against theoretical solution of flow between parallel plates and is in excellent agreement. Also, the immersed boundary method with both the momentum forcing and mass source/sink is found to give realistic velocity profiles and recirculation eddies for the studied flow over forward-backward facing step demonstrating the accuracy of the method. It is to be noted that the existence of flow separation and subsequent reattachment caused by a sudden expansion or compression in flow geometry, such as forward-backward facing steps greatly influences the mechanism of heat transfer. On completion of this numerical study, it may be concluded that immersed boundary method is a robust way of determining flow field with forward-backward facing step and can rival experimental and laboratory study. Generally for all obstructions, the velocities are very small in the recirculation zone compared to the velocity of the mean flow. Hence, the separation surface is submitted to a strong shear. In order to characterize the topology of the separated zone, measurements of the reattachment length in the stream wise direction may be captured.

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