During the past decade, much progress has been made in developing computational
techniques for predicting flow and heat transfer fields. The accuracy
and efficiency of these methods can be affected by factors such as flux
treatment, boundary conditions and the grid type. Many existing methods
have been developed to solve the compressible flow equation within transonic
Mach numbers. However, a lot of applied problems are inherently incompressible
and must be treated appropriately. Even with the high interest in incompressible
flows, the incompressibility requirement has always been an obstacle in
solving the flow equation in a straightforward manner. Since there is
no time-evolution term in the continuity equation, the standard time-discretizing
schemes developed for solving the compressible Navier-Stokes equations
can not be applied directly and the continuity equation imposes a constraint
which the momentum equations have to satisfy.
One of the methods which is used for incompressible flows is called pressure
correction method. This method uses a Poisson equation for the pressure
field (Tsui and Pan, 2006). The other known method is stream function-vorticity
which calculates the values of the stream function and vorticity and determines
the velocity and pressure fields afterwards (Anderson et al., 2006).
With the progress in recent years of the compressible flow schemes, these
schemes have been considered for use with incompressible flows, simply
by lowering the Mach number to minimize the compressibility effects. Unfortunately,
as Mach number is decreased toward incompressible limit, the performance
of compressible methods, in terms of both convergence rate and accuracy
suffers greatly. Guillard and Viozart (1999) have identified that, in
the low Mach number limit, the discretized solution of the compressible
fluid flow equations may fail to provide an accurate approximation to
the incompressible equations.
To overcome the difficulties associated with the use of compressible
methods, excellent progress has been made in applying Artificial Compressibility
Method (ACM) to incompressible flows. ACM, a way of extending the use
of compressible flow schemes for near zero Mach number flows, was first
introduced by Chorin (1967). This method introduces a pseudo time-derivative
of pressure into the continuity equation. This pseudo term changes the
mathematical character of the continuity equation from elliptic to hyperbolic.
This enables the system of equations to be solved with a variety of schemes
developed for compressible flow equations. The method has been successfully
used for both steady and unsteady flows. Tai et al. (2004) developed
a method to solve Navier-Stokes equations, for velocity field, based on
the artificial compressibility method in finite-volume discretization.
Tang and Sotiropoulos (2007) applied the ACM to unsteady problems with
a kind of fractional time stepping procedure on non-staggered grids. Some
researchers have applied ACM in conjunction with different upwind-differencing
schemes and solution techniques to solve steady-state as well as unsteady
incompressible problems. The upwind-differencing schemes that have been
used include the Monotone Upstream-centered Scheme for Conservation Laws
(MUSCL) (Rahman and Siikonen, 2001), Total Variation Diminishing (TVD)
(Choi et al., 2007), Essentially Non-Osilatiory (ENO) (Cheng and
Shu, 2007) and Weighted Essentially Non-oscillatory (WENO) (Yang et
al., 1998; Choi et al., 2007) schemes. Kao and Yang (2007)
have used a segregated finite-difference scheme based on ACM to solve
velocity field of shear-driven cavity. Bassi et al. (2006) have
used Discontinuous Galerkin (DG) methods in conjunction with (ACM) to
solve incompressible Navier-Stokes equations. As Madsen and Schaffer (2006)
have claimed, the time marching approach used in this study can predict
both steady and unsteady flows, but this work will focus on steady condition.
The purpose of the current study is the development of characteristic-based
methods for velocity and temperature fields of incompressible flows using
methods used for solving compressible flow equations. In the new method
artificial compressibility concept was successfully used. To calculate
convective fluxes Riemann solver of Roe was used. The required coefficients,
for both velocity and temperature fields of artificial compressible flows,
were derived. For time-marching, 5th order Runge-Kuta algorithm was utilized
because of its wide range of stability. The method was examined by solving
velocity and temperature fields over circular cylinder and lid driven
cavity and comparing the results by those available in literature. As
authors know there is no study in literature which uses this approach
to solve both velocity and temperature fields of incompressible flows.
Other investigators have used the mentioned characteristic-based method
for compressible flows and usually their study were restricted to velocity
MATERIALS AND METHODS
Governing equations: The primitive form of the incompressible
Navier-Stokes equations in Finite-volume form with artificial compressibility
is given as:
Equation 1 has been non-dimensionalized by the following
Convective flux treatment: At the present time, various flux treatments
are in use. In this study Riemann solver of Roe was applied to calculate
fluxes at cell boundaries. By the use of ACM, the governing equations
took a hyperbolic dominated nature, therefore the application of characteristic
based wave propagation models became possible. Riemann solver of Roe had
been originally developed to estimate the fluxes of compressible Euler
equations (Roe, 1997) but in current study this solver was applied to
the artificial compressible system of Navier-Stokes equations. It can
be shown that in this method fluxes at cell boundary can be written as:
where, NF is the flux vector normal to the grid boundaries, UR
and UL are the values of variables at the right and left side
of cell boundaries and
where, Λ is a diagonal matrix whose elements are the eigen-values
of flux jacobian matrix A and are given by
λ1 = N
λ2 = N
λ3 = N+ c
λ4 = N-c
R and L are right and left eigenvectors of matrix A, respectively. These
matrices for the primitive variables, in the presence of artificial compressibility
factor, has been derived as follows:
where, φ is the shear velocity:
This results lead to the matrix |A|, whose components are:
All the components were calculated using the average of variables, as
The order of accuracy of the scheme depends on choosing UR
and UL values. Assigning cell-center values leads to a first
order accuracy, eq. 14, but using a kind of interpolation
(Drikakis and Rider, 2004) will result in second order of accuracy, eq.
Viscous fluxes: The viscous fluxes discretization is straightforward
and uses 2nd-order averaging. The right hand-side of eq.
1 is approximated by:
In the calculation of ∂ U/ ∂ x and ∂ U/ ∂ y (Fig.
1), secondary cells are employed.
||Approximation of viscous fluxes on secondary cells
∂ U/ ∂ x is approximated as follows:
∂U/ ∂ y is calculated in a similar manner. In which s` is the
secondary cell area.
Time discretization procedure: For time discretization, 5th-order Runge-Kuta
time stepping scheme, because of its wide range of stability (stability margin
of CFL = 4 for the linear advective Eq.) and high accuracy, was utilized. This
allowed that larger time steps to be chosen, hence allowing a faster convergence
to a steady state condition. The solution is updated at consecutive time steps
as shown in eq. 19. The proposed characteristic based method
was able to dampen oscillations, so no oscillation eliminating method was required.
Stability requirement imposes a restriction on the time steps (Chorin,
1967). To have a stable solution time steps should satisfy the following
conditions. By the use of this relation a kind of local time stepping
Boundary condition: Consistent boundary treatment ensures the
disturbance dissipation in discretized domain without reflection. At the
solid boundary a condition of zero mass and energy flux through the surface
was prescribed by setting the fluxes corresponding to these faces equal
to zero. Pressure on the solid surface was found using a kind of interpolation.
Temperature on the solid surface was assumed to be constant. This technique
only permits a flux of pressure term of momentum equation through solid
boundary. At the inlet boundary pressure was extrapolated from interior
domain, velocity and temperature was set equal to the free-stream values.
At the outlet boundary the pressure was fixed, remaining variables were
interpolated from interior domain.
Grid features: An algebric method was used for grid generation
and the grid was clustered near solid boundary. Grid independency is achieved
in a 120x120 resolution.
RESULTS AND DISCUSSION
Numerical results: The method was verified by solving flow over
a circular cylinder and lid driven cavity. For this flow Pr = 0.71 and
CFL = 0.6 was considered. In Fig. 2 Local Nusselt number
around circular cylinder is presented and is compared with results of
other investigators (Chun and Boehm, 1989). The maximum Nusselt number
as expected occurs in the first stagnation point of the cylinder. As boundary
layer develops, thermal resistance increases and local Nusselt number
reduces. The minimum occurs in separation point.
||Local Nusselt No. over circular cylinder
||Comparing average Nusselt No. with emprical relations
In Fig. 3 average Nusslet number against Reynolds Number
is compared with empirical relations. Equation 21-23
are proposed, respectively, by Fand, Knudsen and Katz and Churchill and
Bernstein (Holman, 2002).
C and N coefficients for above relation in the range of 40< Re<
4000 are respectively, 0.683 and 0.466.
The eq. 22 proposed by Fand, for 10-1<10<105,
is simple and has good accuracy. Equation 24 is a little
more complex but covers higher Reynolds numbers.
In Fig. 4 streamlines in Re = 200 and Re = 500 are
presented. Vorticity region behind the cylinder and unsteady nature of
flow is obvious.
It is shown in Fig. 5 that the pressure coefficient
at Re = 40 obtained using the proposed method agrees well with that of
Choi et al. (2007). In Fig. 6 streamlines for
lid driven cavity is compared with the results of Malan et al.
(2002) at Re = 1000, 5000.
|| Streamlines around circular cylinder
||Comparison of wall pressure coefficient along cylinder
surface at Re = 40
||Stream lines for lid driven cavity at Re = 1000, 5000,
a and b current study, c and d (Malan et al., 2002)
||Comparison the convergence rate of proposed method and
averaging method, Re =150
Error norm is defined as:
In the present study a characteristic based method (Riemann solver of
Roe) was applied to velocity and temperature fields of incompressible
flows. The method was examined by applying it to flow over circular cylinder
and shear driven cavity. Artificial compressibility concept made it possible
to use the Riemann solver for incompressible flows which is usually used
for compressible flows. This method was successful in predicting flows
in different Reynolds Numbers. It was able to dampen oscillations, thus
generating a stable behavior. This kind of flux treatment relates the
physical behavior of flow to the mathematics. The flux treatment considers
the Pseudo acoustic wave propagation in the computational domain. This
was made possible by reconstruction of fluxes and their corresponding
eigenvectors. The comparison of convergence rate of the proposed method
and averaging method showed a noticeable reduction in iteration steps.