D
Journal of Communication and Computer 9 (2012) 1104-1109
DAVID
PUBLISHING
On Turbulent Flow Near a Wall
Sabah Tamimi
College of Computing, Al Ghurair University, Dubai 37374, United Arab Emirates
Received: June 12, 2012 / Accepted: June 13, 2012 / Published: October 31, 2012.
Abstract: A finite element method is advocated for representing equations governing confined turbulent flow. In particular, a
technique for the effective mapping of regions close to the solid boundaries has been adopted for predicting the distribution of the
pertinent variables in this zone in a smooth straight channel. In the present work, the validity of the imposed technique has been
tested when two types of flow were examined, pressure flow and the combination of pressure and Coquette flow with a one equation
model used to depict the turbulent viscosity.
Key words: Combination of pressure and couette flow, wall element technique, FEM.
1. Introduction
The N-S (navier-stokes) equations governing the
fluid motion are known to apply in a wide range of
applied computer science as well as engineering
disciplines.
Due to the complexity of these equations an
analytical solution is intractable and the last three
decades, attention has been focused on the numerical
simulation of flow processes, the so-called CFD
(computational fluid dynamics). This has been
developed and used with confidence to solve a large
range of flow problems. Indeed, under conditions
where experimentation is extremely difficult, CFD
may be the only methodology available. Numerous
theoretical and experimental works are available on
laminar flow [1-4], but in case of turbulent flow are
still few. Since it has not been possible to obtain exact
analytical solutions to such flows, an accurate
numerical approach would be very beneficial to
researchers. The FEM (finite element method) is one
of the numerical methods that has recently emerged as
a powerful tool for solving the N-S equations. Within
the computational domain, the finite element method
Corresponding author: Sabah Tamimi, Ph.D., associate
professor, research fields: CFD, computer graphics, modeling
and simulation. E-mail: Sabah@agu.ac.ae.
is used to discretise the equations governing fluid
motion, namely continuity and the N-S equation. For
present purposes the analysis is approached via a time
averaged form of N-S equation with a spatially
varying viscosity. For closure additional equations are
required in order to evaluate local values of the
turbulence kinetic energy, k, and Prandtl’s mixing
length. The additional equations must, therefore,
depict the variation of both k and the mixing length. In
the present work, the one equation model of
turbulence is used, and a transport equation is derived
which can be used for evaluation of k. The aim of the
present work is to report a finite element based
solution technique for general steady state, two
dimensional, incompressible, confined turbulent flow.
Particular attention is given to the important aspect
of studying the flow behaviour in region close to the
solid wall which known as ‘N.W.Z’ (near wall zone)
of confined flow. It very important that the transfer of
mass and momentum within this zone is modeled
accurately in order to obtain correct overall
predictions particularly the transfer of shear from the
solid wall, with an associated large variation in
velocity and kinetic energy. If a conversational finite
element (i.e. two dimensional elements up to the wall)
is used to model the near wall zone then significant
On Turbulent Flow Near a Wall
grid refinement would be required which would be
very costly in both computer storage and CPU time. In
order to avoid such excessive refinement, several
solution techniques have been suggested [5-7]. The
more common approach is to terminate the discretised
domain (main domain) at some small distance away
from the wall ‘near wall zone’, where the gradients of
the independent variables are relatively small. One of
them is the so-called universal laws.
In present work, one dimensional finite element
technique (i.e. one dimensional element in one
direction normal to the solid wall) has been used. The
validity of this wall element technique has been tested
and proved in the previous work [8] when pressure
flow was considered. In this paper, wall element
technique has proved again its validity when a
combination of pressure and Coquette flow was
considered.
2. Governing Equations
The current investigation relates to steady—state
incompressible two dimensional turbulent flow of a
Newtonian viscous fluid with no body forces acting.
For such a situation, the N-S equations associated with
this type are which are,
p
 ui

=+
x j
x j
xi
  u u j  
  (1)
  e  i 

   x j  x i  
where i,j= 1, 2. ui, p are the time-averaged velocities
uj
and pressure respectively,  is the fluid density, e is
the effective viscosity which is given by e =  + t, 
and t are the molecular viscosity and turbulent
viscosity, respectively. The flow field must satisfy the
continuity equation, which may be written as：
 ui
= 0
x i
(2)
Eqs. (1) and (2) can not be solved unless a
turbulence closure model can be provided to evaluate
the turbulent contribution to  e . The simplest model
is via an algebraic formula [9] which has limited
application and therefore this model is not adopted in
1105
the present work, but an alternative Prandtlkolmogorov [10] model is used in which,
 t C   k 1 / 21
(3)
1 is the length scale of turbulence which obtained
by 2.5*1m , where 1m is the mixing length based on the
prandtl hypothesis which has been specified
algebraically for the present purposes as 0.4 times the
normal distance from the nearest wall surface, C is a
constant and k is the time-averaged turbulence kinetic
energy. The t given by Eq. (3) requires that k to be
known. This can be evaluated via a further transport
equation given by:
uj
k   t  k  ui ui uj 
     t
  B (4)
xj xj  k  xj  xj xj xi 
where B = C D  k 3 / 2 /1  ,  t /  k is the turbulent
diffusion coefficient,  k is the turbulent prandtl and
CD is a constant.
The turbulence model based on equations (1-2) and
(4) is called the one-equation (k-l) model.
Within the main domain the above governing
equations have been discretised by using the standard
finite element method [11-12] and Galerking weight
residual approach is adopted to solve the discretised
equations using quadratic 8-nodded elements to define
the variations in velocities and turbulent kinetic
energy, and linear 4-nodded elements used for the
pressure. Greens theorem is used then to reduce the
order of the equations to unity resulting in a “weak
formulation” which resulted in non-linear equations
matrix which is solved by either a coupled or an
uncoupled method.
3. Near Wall Zone Modeling Techniques
Within the near wall zone different techniques were
used (Fig. 1), these are as follows,
(1) Conventional finite elements (i.e.2-D elements
up to the wall) are used to discretise the N.W.Z. and
the variable values, following analysis, are used as
reference data. However an excessive mesh
On Turbulent Flow Near a Wall
1106
refinement was needed which is expensive in
computer time and memory.
(2) In order to avoid such excessive refinement,
semiempirical equations, known as “Wall Functions”
or the so-called “Universal Laws” [13], are used to
bridge from a solid boundary to the main domain.
(3) In the present work, a finite elements technique
has been adopted, using one-dimensional (3-nodded
elements) normal to the wall (Fig. 2).
4. Boundary Conditions
In the present work, two types of fully developed
turbulent flow are considered. These are pressure (i.e.
both walls of the channel are fixed) and pressure plus
Couette flow pressure (i.e. one of the walls is moving
and the other one is fixed), in our case the upper wall
is moving and the lower wall is fixed. In both, fully
developed Dirichlet conditions are assumed on all
variables upstream. No slip condition were imposed
on solid boundaries and tractions updated downstream.
Tractions are given by,
 x  p 
1
x 
2
 e  u1 


  x1 
x1—parallel to walls
 e  u2 u1 

 x2—normal to walls

  x1 x2 
5. Results and Discussion
Full developed turbulent flow was considered in a
parallel-sided duct of width D, which is taken as 1.0 in
the present work, and length L. Compatible fully
developed velocity and kinetic energy profiles were
imposed as initial upstream values. These profiles
were obtained by using the outlet values form each
iteration as new approximation to the values at the
inlet until a converged condition is satisfied. Different
Reynolds number based upon the width of the channel
of 70.000, 50.000 and 12.000 were considered.
Fig. 1
Boundary conditions when the mesh is terminated at small distance away from the wall.
Fig. 2
One-dimensional elements in one-direction normal to the wall used in the N.W.Z.
On Turbulent Flow Near a Wall
The first example was concerned with an analysis
of pressure flow where both walls of the duct were
fixed. Fig. 3 clearly shows that the velocity values
obtained by universal profiles have some discrepancy
from those obtained from the advocated wall element
technique (i.e. one-dimensional element in one
direction). Again, the results obtained from the
adoption of the presently advocated technique exhibits
excellent agreement with the correct solution which
resulted from the complete mapping (i.e.
two-dimensional element up to the wall) as shown in
Fig. 4. These are, superior to those obtained using
universal laws. Fig. 5 shows excellent agreement
between the imposed technique and experimental
results [14].
Once more, the “correct” values are remarkably
close to those obtained from the proposed technique,
1107
as shown in Fig. 6 which refers to the kinetic energy
profile. Obviously the results obtained from the
adoption of 1-D elements in one direction is still the
most advantageous owing to the number of elements
used in the near wall zone.
The second example was concerned with an
analysis of combining pressure and Couette flow, with
the lower surface stationary and upper surface moving
at a constant speed. Fully developed turbulent velocity
profiles and turbulent kinetic energy distribution for a
Reynolds number of 70.000 were obtained and
presented in Figs. 7 and 8, respectively, these show
comparisons with universal laws and experimental
results [14]. Once more, the results obtained from the
adoption of the wall element technique are significantly
better than those obtained using the universal laws, and
compare favourable with experimental results.
Fig. 3
Turbulent velocity profiles for fully-developed flow, at 8D downstream, L = 8D, Re = 50.000.
Fig. 4
Turbulent velocity profiles for fully-developed flow, at 8D downstream, L = 8D, Re = 12.000.
1108
On Turbulent Flow Near a Wall
Fig. 5
Turbulent velocity profiles for fully-developed flow, at 8D downstream, L = 8D, Re = 50.000.
Fig. 6
Kinetic energy profiles for fully-developed turbulent flow, at 8D downstream, L = 8D, Re = 12.000.
Fig. 7 Turbulent velocity profiles for fully-developed for turbulent flow with fixed lower surface and moving upper surface,
at 8D downstream, L = 8D, Re =70.000.
On Turbulent Flow Near a Wall
1109
Fig. 8 Turbulent fully-developed kinetic energy profiles for turbulent flow with fixed lower surface and moving upper
surface, at 8D downstream, L = 8D, Re = 70.000.
6. Conclusions
The utilization of universal laws is not valid since
these laws are only really applicable for certain one
dimensional flow regimes. Also, the use of 2-D
elements up to the wall is not economically viable.
Therefore to avoid such an excessive refinement, these
methods have been replaced by introducing a finite
element wall element technique which has shown
more advantages comparing to these methods. The
validation of the wall element technique has been
proved again its validity with case of the combination
of pressure and couette flow. Therefore, this technique
can be used with confidence for fully-developed
turbulent flow.
[5]
[6]
[7]
[8]
[9]
[10]
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