jangReport_laminarchannel

8/10/2019 jangReport_laminarchannel

1/5

Proceedings of

, ,

BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATIONIN 2D CHANNEL FLOW

Yunho JangDepartment of Mechanical and Industrial Engineering

University of MassachusettsAmherst, MA 01002

Email: [email protected]

ABSTRACT

The work on this theme will comprise a boundary layer anal-

ysis in channel flow. Here we will be looking at both the lami-

nar and turbulent case of incompressible flow within the pres-

ence of shear stress and vorticity. This study for both cases is

a very important concept to understand for boundary layers in

channel flows. To accomplish this study, we used the Finite El-

ement Method and Finite Volume Method, and compared with

Direct Numerical Simulation data for channel flow. Boundary

layer simulations of fully developed laminar and turbulent chan-nel flow at two Reynolds numbers up to Re = 590 are reported.

INTRODUCTION

The plane channel, which is also called plane Poiseuille flow

or duct flow, is a canonical configuration for studying internal

flows. Understanding the structure of channel flow is obviously

of great engineering interest since this can be applied in many

applications. This flow is obviously a Newtonian fluid, so that

the important boundary problems are raised. To study the plane

channel flow, we need to understand the behavior of flow in

boundary layers for both laminar and turbulence flows.For the laminar channel flow, we know the solutions since

we could calculate it analytically, but in turbulent case, we can

not get an analytical turbulent solution since turbulence is more

complex, high Reynolds number is applied, and becomes unsta-

ble. Moreover, the reason why turbulence is more complex is the

boundary layer starts off laminar, but at some critical Reynolds

number, it becomes unstable to disturbance, e.g. noise, vibra-

tion, surface, and so on. Therefore, we will discuss the boundary

layer in detail in this study. However, we could also approach the

turbulent channel flow solution diffusion term, and understand

the phenomena of boundary layer in turbulent channel flow with

modeling.

The Finite Element Method is widely used for numerica

analysis because it is not only accurate but also provides complex

mesh grids. Therefore, we could solve many other application

of channel flows with Finite Element Method. In this study, we

solve two cases of channel flow. First one is the laminar case

(Re= 90) of channel flow. We can obtain analytical solutionsand compare with results from Finite Element Method (Ansys

and Finite Volume Method (Fluent). For the next step, we solve

the turbulence case (Re= 590), and compare with results fromthe FVM and MKM1 DNS (Direct Numerical Simulation) data

We are interested in understanding the phenomena of the dif-

ferences between a laminar and a turbulent flow, and also how

the finite element method can predict well the channel flow with

comparing to other data. For turbulent channel flow, thek model has been employed.

Formulation

We begin with the equations for two dimensional steady con

tinuity and Navier-Stokes equations.

u

x+v

y=0 (1

1Moser,R.D.,Kim,J.and Mansour,N,N(1999)

1 Copyright c by ASME


2/5

uu

x+ v

u

y=

1

p

x+(

2u

x2+

2u

y2) (2)

uvx

+ vvy

=1py

+(2vx2

+ 2vy2

) (3)

Laminar channel flowFor laminar channel flow, the no-slip boundary condition has

been employed and we can apply the conservation of mass and

momentum, then we can get the solution for the horizontal veloc-

ity, average velocity, vorticity, and the shear stress at the bottom

wall;

u(y) =umax[14(y

h)],umax=

h2

8

d p

dx(4)

um=2

3umax (5)

yx=du

dy=8umax

y

h2 (6)

yx= dudy

=8umax yh2

(7)

where h,um, and umax are height of channel, mean velocity, and

maximum of horizontal velocity, respectively.

Re=u

(8)

u=

w

(9)

where and u are boundary layer thickness and wall shear ve-locity, respectively;

We use constant dynamic viscosity and density ( = 0.01111and = 1) for incompressible flow so that we have constant kine-matic viscosity (=/=0.01111). For this laminar case, theheight of channel is 2m, length of channel is 100m, the grid of

domain is 60*200, and Reynolds number based on u is 90.

From equation (5) and (6), we know the maximum horizon-

tal velocity is 45m/s, and the mean velocity is 30m/s. Moreover

we can calculate following quantities with this analytical solu-

tions.

Rem=um

2

(10

Rec=uc

(11

whereuc is centerline velocity, andCf is skin fraction based on

w and um.

Cf= w

12um2

(12

Cfis skin fraction based onw and umWe also use equations for displacement thickness () and momentum thickness ().

=

0

(1 u

uc)dy (13

=

0

[(1 u

uc)

u

uc]dy (14

Turbulent channel flow

For turbulent channel flow, the k model has been usedand Re is 590, and we introduce the standard kmodel whichis modeling the turbulent viscosity from the transport equation

The turbulent (or eddy) viscosity, t, is computed by combining

k and as follows,

t= Ck2

(15

whereC is a constant. In this study we follow default values o

the model constants, C1,C2,C, and , which have been determined from experiments with air and water for fundamental

turbulent flows and they have been found to work fairly well fo

a wide range of wall-bounded and free shear flows. The mode

constants are,

C1=1.44, C2=1.92, C=0.09,=1.3,k=1.0



3/5

For usingk model, it is assumed that the flow is fully turbu-lent, and the effects of molecular viscosity are negligible. The

standardk model is therefore valid only for fully turbulentflows.

The standard wall function has been used for wall boundary

treatment, resulting in,

u+ =1

ln(Ey+) (16)

where

u+ UpC

1/4 k

1/2p

w/ (17)

y+ C

1/4 k

1/2p yp

(18)

and

= Von Karman constant (= 0.42)E= empirical constant (= 9.81)

Up = mean velocity of the fluid at point p

kp = turbulence kinetic energy at point p

yp = distance from point p to the wall

= dynamic viscosity of the fluid

In turbulent channel flow, the kinematic viscosity and mean

velocity are reduced to get Re = 590 ( = 0.001695, um =18.4539).

Results

Laminar channel flow

As mentioned before in this chapter, the data from Finite El-

ement Method will be compared with an analytical solution and

the data from Finite Volume Method. From the analytical so-

lution for laminar channel flow, we have predictable maximum

velocity, shear stress, vorticity, and so on. As shown in Table 1,

the quantities from the FEA and FVM simulations are very sim-

ilar with those from analytical solution.Figure 1 shows velocity profiles from FVM and FEA. The

mean velocity is 30 m/s in this case, and as shown this is the

maximum velocity which is centerline velocity is almost 45 m/s

(analytical solution) in both cases. Figure 2 and Figure 3 show

the behaviors of shear stress and vorticity in channel flow. The

shear stress and vorticity are at a maximum at the wall, then grad-

ually decrease with distance. Those values are also very similar

with those of analytical solution.

Table 1. COMMONLY COMPUTED QUANTITIES

Method umax(m/s) w (pascal) w(s1)

Analytic solution 45 1.0 -90

FVM 44.9313 0.987 -88.8

FEA 44.248 0.935 -84.144

Method Cf Cf o

Analytic solution 0.00222 0.000987 0.333

FVM 0.00219 0.000974 0.356

FEA 0.00207 0.000923 0.338

Method Rem Rec

Analytic solution 0.1333 5400 4050

FVM 0.1343 5400 4043

FEA 0.1338 5400 3982

Figure 1. VELOCITY PROFILES IN FULLY DEVELOPED LAMINAR

CHANNEL FLOW AT X/L = 1, Re = 90

Turbulent channel flowIn this part, we will discuss the results of turbulent channel

flow. As shown before, in this case Re is 590, and viscosity (= 0.001695) has been reduced to match this Re.

The velocity profiles (Figure 4) are very different from the

laminar channel flow. As shown, the velocities in middle of chan

nel level-off due to the behavior of turbulence viscosity. For FEA

data, the velocities in middle of channel are smaller than those of

FVM and Direct Numerical Simulation (DNS) data, however, as



4/5

Figure 2. SHEAR STRESS PROFILES IN FULLY DEVELOPED LAMI-NAR CHANNEL FLOW AT X/L = 1,Re = 90

Figure 3. VORTICITY PROFILES IN FULLY DEVELOPED LAMINAR

CHANNEL FLOW AT X/L = 1, Re = 90

we approach the wall, they are much similar to DNS data. Fig-ure 5 and Figure 6 show the shear stress and vorticity profiles

in turbulent channel flow. In this picture, turbulent channel flow

is not dominated by the shear stress and vorticity. At the wall,

vorticity is increasing compared with laminar channel flow.

Figure 7 shows the wall law plot for turbulent boundary lay-

ers with three sets of data. The logarithmic law for mean velocity

is known to be valid fory+ >about 30 to 60. In our case, the log-law is employed wheny+ >11.225. When the mesh is such that

Figure 4. VELOCITY PROFILES IN FULLY DEVELOPED TURBULENTCHANNEL FLOW AT X/L = 1, Re = 590

Figure 5. SHEAR STRESS PROFILES IN FULLY DEVELOPED TUR

BULENT CHANNEL FLOW AT X/L = 1, Re = 590

y+


5/5

Figure 6. VORTICITY PROFILES IN FULLY DEVELOPED TURBU-LENT CHANNEL FLOW AT X/L = 1,Re = 590

Figure 7. Mean velocity profiles in fully developed turbulent channel flow

at X/L = 1,Re = 590

CONCLUSIONWe simulated both laminar and turbulent channel flow using

FEA, and compared with data from FVM and DNS. As shown

in results, in the laminar case the data from FEA and FVM are

quite similar to the analytical solution; and velocity, shear stress,

and vorticity profiles are close to each other.

Therefore, we can say that the simulations of laminar flow

which is dominated by viscosity are well predictable in both Fi-

nite Volume Method and Finite Element Method, and it is possi-

ble to apply different simulations to the boundary layer problems

In the turbulent channel flow case, there are some errors

since we are using turbulent model (k ) which is not an ex-act solution. However, the data from FEA and FVM are believ-

able and still going on right track. In order to understand abou

the turbulent flow, we need to develop more exact model for thisflow.

REFERENCES[1] Moser,R.D., Kim,J. and Mansour,N.N., Direct Numerica

Simulation of Turbulent Channel Flow up to Re = 590

Phys.Fluid, Vol 11,No4, pp 943-945. 1999.

[2] Pope, Stephen B., Book:Turbulent Flows, Cambridge Uni

versity Press, New York, 2000.

[3] Schetz, Joseph A., Book: Boundary Layer Analysis

Prentice-Hall,Inc., New Jersey, 1993.

[4] Wilcox, David C., Book: Basic Fluid Mechanics, DCW In

dustries,Inc., California, 2000.

[5] Fluent 6.0 Manual.

[6] Ansys 5.7 Manual.


Documents

jangReport_laminarchannel