PREDICTION OF TURBULENCE FLOW WITH EFFECTS
OF STREAMLINE CURVATURE
Wei Wen Li1, 2 Liu Yu Ling1
Li Jian Zhong1
(1 Xi’an University of
Technology, Xi’an 710048, China)
(2
Dalian University of Technology, Dalian 116024, China)
Abstract:
A mathematical model for predicting turbulence flows with effects of streamline
curvature under orthogonal curvilinear coordinates is developed in this paper
.The SIMPLEC solution procedure has been used for the transformed governing
equations in the transformed
domain. The Loci of flow reversal of the seperated flows over a backward facing
step are employed to test the capability of the proposed turbulence model in
capturing the effects of local curvature .The comparision between the computed
results and experimental data shows a satisfactory agreement .
Keywords:
orthogonal curvilinear coordinates, mathematical model, effects of streamline
curvature
1 INTRODUCTION
The tremendous improvement of computer
capabilities, in memory and speed, has enabled accurate numerical predictions of
turbulent flows. Due to the closure problem of the governing equations for
turbulence flows, numerous turbulence models have been proposed. The
eddy-viscosity type of turbulence closure modeling has demonstrated a variety of
good numerical predictions both qualitatively and quantitatively. Among them,
the k-
model is the most widely employed isotropic two-equation model. It has been
extensively applied to different turbulent flow problems. However, the standard
k-
model appears to be insufficient in predicting the complex turbulence shear
layers, such as flows subjected to curvature and rotation. The main reason is
that the streamline curvature produces unexpectedly large changes in boundary
layer properties and that the eddy-viscosity for standard k-
model is isotropic. So several researchers have discussed the sensitivity of
turbulence flow characteristics to even small amounts of mean streamline
curvature. For example, in the study by Kreith [1] and in subsequent
investigations by Thomann[2] and Mayle et al [3] , it has been shown that the
best flux through the concave wall of a curved channel can be up to 33% larger,
and through the convex wall 15%smaller, relative to that through the walls of a
straight channel. Therefore, numerous models have been proposed in the last two
decades to account for the effects of streamline curvature, such as [4-9]etc.
In order to combine the simplicity (i.e., easily
adopted into other programs or models ), generality (suitable for different
geometries ), physical rationale, and efficiency (less computing time), the
present study applies the curvature correction method by Launder et al. And
Sharma[7] in the two-equation
turbulence model under orthogonal curvilinear coordinates. Thus, a mathematical
model for prediction of turbulence flow with effects of streamline curvature
under orthogonal curvilinear coordinates has been established and will be widely
used in the prediction of complex turbulence flows in hydraulic engineering.
2 MATHEMATICAL MODEL
2.1
Generation of orthogonal curvilinear grids
Generating a grid in an arbitrary physical
domain involves a coordinate transformation from the physical plane (x, y) to
the computational plane (
) (see Fig.1). This is done here by solving a system of Poisson equation
(1a)
(1b)
where
and
.
In Eq. (1), how to determine the functions of P
and Q is often difficult. Wei Wen Li [11] has proposed a new method to determine
them more efficiently with which a desired boundary-fitted curvilinear
coordinate grid can be automatically generated. The details of the method are
referred to Ref.[11]. The expressions of the functions P and Q are given below:
(2a)
(2b)
where
Now considering the functions P and Q in (2),
and solving Eq.(1),we can generate a boundary-fitted orthogonal curvilinear
system. Eq.(1) are rewritten in finite difference scheme and then solved by the
ADI method. If the points on the boundaries are unreasonably selected, the
generated grids may be less orthogonal. Therefore, we make the points slip along
the boundaries under the condition,
. Further more, all subsequent hydrodynamic computations are performed in the
coordinates(
).
2.2 Governing equations
in transformed plane
The governing equations for turbulent flows are
the Reynolds-averaged Navier-Stokes equations. In the equations, the Reynolds
tress (
) or turbulent stress appears. Therefore, additional equations are needed to
solve the system of equations. The
model, proposed by launder and Spalding (1972), is the widest applied
two-equation turbulent closure model, and in the
model, the Reynolds stresses have been modeled according to the Boussinesq
assumption which relates the stresses to velocity gradients through a turbulent
viscosity.
For two dimensional, time-averaged,
incompressible turbulent flows, the governing equations can be written in
orthogonal curvilinear coordinates in the following general form as
(3)
where U and V are the
velocity components in the
and
directions, respectively;
represents any dependent variable
of interest;
is the source term in the
coordinate system;
is the effective viscosity. If the conditions
and
are satisfied, Eq. (3) represents the continuity equation.
and
for each of the transport
quantities (U, V, k,
) are as follows, respectively
(
momentum equation)
(4a)
(4b)
momentum equation)
(5a)
(5b)
(k transportation equation)
(6a)
(6b)
(
transportation equation)
(7a)
where
and
are gravity acceleration in
and
directions, respectively;
and
are empirical constants and have the values of 1.0,1.44,1.92,and 1.3,
respectively;
is the laminar viscosity;
the turbulent production term;
the turbulent viscosity. The expressions of
and
are as follows, respectively
(8)
(9)
where
is the modeling constant in the turbulent viscosity formulation, and has the
value of 0.09, as shown in Eq.(9).
As stated above, this is the standard
two-equation
model in an orthogonal curvilinear coordinate system. The
model can not predict complex turbulent flows. A few suggestions for modifying
the
model have been published, which aimed at the
-equation. Launder and Sharma[7] selected a special form of the gradient
Richardson number, involving a typical turbulence frequency to characterize the
influence of the curvature:
(10)
where U is the velocity in the streamline direction, r is the radius of
curvature of the streamline.
For the modified destruction term in the
equation, they proposed
(11)
and assigned a value of
0.20 to the constant
. This modification gave good predications for data on a curved surface [7].
This basic idea has been adopted in this paper.
2.3 Discretization
scheme
A
staggered grid system is used where the control volumes for U
and V are Centred on the faces of the control volumes for the scalar
variables,
and
, the pressure nodes are located at the center of the continuity control volume,
which is known to surpress the wiggles or the wiggles or the checkboard patterns
of the pressure [10], as shown in Fig.1.
(a) Physical plane
(b) Transformed plane
Fig.
1 Control volume grid system
2.4 Boundary conditions
(1) Inlet plane: U, V, k and
are specified.
(2) Exit plane:
are satisfied.
(3) Body surface: Wall-functions are used for
turbulent kinetic energy
and its rate of dissipation
; and no slipping condition is used for velocity.
3 RESULTS AND
DISCUSSIONS
The flow over a backward-facing step has been
used to test the proposed model for the flow with local curvature effect. The
computed region is shown in Fig.2.
Fig.2
Computed region
The comparision is made between the present
model and the standard
model with 71×21
grid points. The flow prediction improvement by the present model is observed
from the locus of flow reversal illustrated in Fig.3. The result indicates that
the present model predicts later flow reattachment than does the standard
model. It appears that the present model can account for the reduction of the
eddy- viscosity by the effect of convex curvature on the primary flow as the
flow separates from the step. The flow separation will also generate the effects
of concave curvature on the secondary flow in the recirculation zone. The effect
of the eddy – viscosity increase in the recirculation zone, thus, is much
smaller than the effects of the eddy- viscosity reduction in the primary flow.
Moreover, since there is no mechanism in the standard
model to simulate the curvature
effects, the predicted convex shear layer exhibits a higher viscosity and
earlier reatchment in the standard
model.
measured;
------s-k-
computed; ——c-k-
computed
Fig.3
Comparision of locus of flow reversal between computed results and experimental
data for different Re
4 CONCLUSIONS
Advantages of this proposed mathematical model
are that the effects of streamline curvature to properties of turbulence flow is
included and the orthogonal curvilinear coordinate grids are used to deal with
the complicated computational region boundaries in the numerical simulation of
complex turbulence flow. The computed examples show that this proposed model has
good stability , convergence and accuracy; and this model will find more
applications in hydraulic engineering.
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