COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Implicit Numerical Scheme Based on SMAC Method for Computing
Unsteady Incompressible Turbulent Flows
Y. X. Zhang1*, Z. L. Li1, B. S. Zhu2, L. Zhang2
1
2
Department of Mechanical and Electronic Engineering, China University of Petroleum-Beijing, Beijing,
102249 China
Department of Thermal Engineering, Tsinghua University, Beijing, 100084 China
Email: [email protected]
Abstract An implicit numerical scheme was developed based on the simplified marker and cell (SMAC)
method for computing the three-dimensional (3-D) unsteady incompressible turbulence Reynoldsaveraging equations in general curvilinear coordinates. The governing equations included the Reynolds
averaged momentum equations with contravariant velocities as the unknown variables, pressure Poisson
equation and k − ε turbulent model equations. The governing equations were discretized in 3-D MAC
staggered grid. To improve the numerical stability of the implicit SMAC scheme, the higher-order high
resolution Chakravarthy-Osher TVD scheme was used to discretize the convective terms in momentum
equations and k − ε equations. The algebraic equations of momentum equations and k − ε equations were
solved by CTDMA method. The algebraic Poisson equations were solved by the Tschebyscheff SLOR
method with alternating computational directions. The unsteady flows through a simplified cascade made
up of NACA65-410 blade at high Reynolds numbers were simulated with the code written by the first
author. The numerical results of the surface pressure coefficient were in satisfactory agreement with the
experimental data. The numerical results indicate that the Reynolds number and the angle of attack were
two primary factors which affected the unsteady characteristics of the flows.
Keywords unsteady incompressible flows, SMAC Method, contravariant velocity, TVD scheme, cascade
INTRODUCTION
The unsteady incompressible flows involve flow separation, recirculation, vortex shedding, turbulence and
so on. It is a formidable task to simulate these practical flows accurately. Ghia et al developed an unsteady
direct solution method using the vorticity and stream function in general curvilinear orthogonal
coordinates [1]. Rao et al presented a primitive variables solution method using Cartesian velocity
components. In this method, the third-order upwind scheme was adopted to discretize the convective
terms [2]. Rosenfeld et al developed a fractional step solution method for determining the 3-D viscous flow
in a generalized coordinates system [3]. Rogers and Kwak extended the artificial compressibility methods
to unsteady incompressible flows by the introduction of the second-order backward-difference scheme and
the Newton iteration with dual-time stepping to restore the time accuracy of the solution [4]. Among these
methods, that with the velocity and pressure as the unknown variables was of great benefit to the numerical
simulation of turbulent flows. The conventional turbulence models could be used directly. The MAC
(marker and cell) method is one of the primitive variables methods for solving incompressible flows. The
SMAC method was developed based on the MAC method [5]. It inherited the two characteristics of the
MAC method including using MAC staggered grid and computing Poisson equation. In this paper, the
authors developed an implicit SMAC scheme in general curvilinear coordinates for unsteady
incompressible flows. To validate the scheme, the unsteady flows of a 2-D cascade were calculated. The
⎯ 306 ⎯
structures of flow fields and the mechanism of unsteady flows were investigated preliminarily.
GOVERNING EQUATIONS
The fundamental equations of unsteady incompressible turbulent flows are Reynolds-averaged N-S
equations and continuity equation of contravariant velocity in general curvilinear coordinates. They can be
written in conservative form as follows
∂ ( JU l )
∂t
+
∂ ( JU iU l )
∂ξi
(
∂ −ui' u 'j
∂ ( h jk Z k )
⎡
⎤
∂
∂p
− u ⋅ ⎢ JU i
∇
=
F
−
Jg
−
+
J
ξ
νε
α
( l )⎥ l
li
lij
ijkl
∂ξi
∂ξi
∂ξi
∂ξ k
⎣
⎦
)
( l = 1, 2,3)
(1)
∂
( JU i ) = 0
∂ξi
(2)
where ε lij is the permutation tensor. The metrics gij and hij are contravariant tensor components and
covariant tensor components, gij = ∇ξi ⋅ ∇ξ j , hij =
J=
∂ ( x, y , z )
∂ (ξ , η , ζ )
∂xk ∂xk
∂ξ ∂ξ
. α ijkl is defined as α ijkl = l k . J is Jacobian,
∂ξi ∂ξ j
∂xi ∂x j
. Fl is inertial force. Z k is contravariant vorticity, Z k = ∇ξi ⋅ Ω , Ω = ∇ × u .
It is necessary to introduce turbulence model to make the equations (1) and (2) closed. In this paper, the
standard k − ε turbulence model is adopted. The k − ε equations in general curvilinear coordinates can be
written as follows
ν ⎞
∂
∂
∂ ⎡⎛
∂k ⎤
( Jk ) + ( JU i k ) = ⎢⎜ν + T ⎟ Jgij ⎥ + J ( P − ε )
σk ⎠
∂t
∂ξi
∂ξi ⎣⎢⎝
∂ξ j ⎥⎦
(3)
⎛
ν ⎞
∂
∂
∂ ⎡⎛
∂ε ⎤
ε
ε2 ⎞
( J ε ) + ( JU i ε ) = ⎢⎜ν + T ⎟ Jgij ⎥ + J ⎜ Cε 1 P − Cε 2 ⎟
∂t
∂ξi
∂ξi ⎢⎣⎝
∂ξ j ⎦⎥
k
k ⎠
σε ⎠
⎝
(4)
Where k , ε , P and ν T are the turbulence kinetic energy, the dissipation rate of k , the production of
k
1
2
and the eddy viscosity respectively. They are defined as k = ui' ui' , ε =
⎛ ∂u ∂u j ⎞ ∂ui
P = νT ⎜ i +
⎜ ∂x j ∂xi ⎟⎟ ∂x j
⎝
⎠
∂ui' ∂ui'
∂xl ∂xl
, ν T = Cμ
k2
ε
,
. The coefficients are Cμ = 0.09 , σ k = 1.0 , Cε 1 = 1.44 , Cε 2 = 1.92 , σ ε = 1.3 .
The usual SMAC scheme for incompressible N-S equations in general curvilinear coordinates is explicit. It
can be calculated directly and easily. However, in the explicit scheme, the time increment Δt is strictly
limited to satisfy the CFL number. To improve the numerical stability and computational efficiency, the
implicit scheme developed for solving the steady flows [6] is extend to unsteady flows. The governing
equations of implicit SMAC scheme for three-dimensional unsteady incompressible turbulent flows can be
written as follows including momentum equations, k equation, ε equation and Poisson equation.
∂ ⎛ h23 ∂ h23 h33 ∂ h22 ⎞ ⎤ ⎪⎫ ⎪⎧ Δt ⎡ ( m ) ∂ 1
∂ ⎛ h13 ∂ h13 h33 ∂ h11 ⎞ ⎤ ⎪⎫
⎪⎧ Δt ⎡ ( m ) ∂ 1
+ν
−
+ν
−
⎨1 + ⎢ JU
⎜
⎟ ⎥ ⎬ ⋅ ⎨1 + ⎢ JV
⎜
⎟⎥ ⎬ ⋅
∂ξ J
∂ξ ⎝ J ∂ξ J
∂η J
∂η ⎝ J ∂η J
2 ⎣
2 ⎣
J ∂ξ J ⎠ ⎦ ⎭⎪ ⎩⎪
J ∂η J ⎠ ⎦ ⎭⎪
⎩⎪
∂ ⎛ h12 ∂ h12 h22 ∂ h11 ⎞ ⎤ ⎪⎫
⎪⎧ Δt ⎡
* m −1
*( m )
( m) ∂ 1
+ν
−
= RHS1 ( )
⎨1 + ⎢ JW
⎜
⎟ ⎥ ⎬ ΔJU
∂ζ J
∂ζ ⎝ J ∂ζ J
2 ⎣
J ∂ζ J ⎠ ⎦ ⎪⎭
⎪⎩
∂ ⎛ h23 ∂ h23 h33 ∂ h22
⎪⎧ Δt ⎡ ( m ) ∂ 1
+ν
−
⎨1 + ⎢ JU
⎜
∂ξ J
∂ξ ⎝ J ∂ξ J
2 ⎣
J ∂ξ J
⎪⎩
⎞ ⎤ ⎪⎫ ⎪⎧ Δt ⎡ ( m ) ∂ 1
∂ ⎛ h13 ∂ h13 h11 ∂ h33 ⎞ ⎤ ⎪⎫
+ν
−
⎟ ⎥ ⎬ ⋅ ⎨1 + ⎢ JV
⎜
⎟⎥ ⎬ ⋅
∂η J
∂η ⎝ J ∂η J
2 ⎣
J ∂η J ⎠ ⎦ ⎪⎭
⎠ ⎦ ⎪⎭ ⎪⎩
∂ ⎛ h12 ∂ h12 h11 ∂ h22 ⎞ ⎤ ⎪⎫
⎪⎧ Δt ⎡
*( m )
* m −1
( m) ∂ 1
+ν
−
= RHS 2 ( )
⎨1 + ⎢ JW
⎜
⎟ ⎥ ⎬ ΔJV
∂
∂
∂
∂
ζ
ζ
ζ
ζ
2
J
J
J
J
J
⎝
⎠ ⎦ ⎭⎪
⎣
⎩⎪
⎯ 307 ⎯
(5)
(6)
∂ ⎛ h23 ∂ h23 h22 ∂ h33 ⎞ ⎤ ⎪⎫ ⎪⎧ Δt ⎡ ( m ) ∂ 1
∂ ⎛ h13 ∂ h13 h11 ∂ h33 ⎞ ⎤ ⎪⎫
⎪⎧ Δt ⎡ ( m ) ∂ 1
+ν
−
+ν
−
⎨1 + ⎢ JU
⎜
⎟ ⎥ ⎬ ⋅ ⎨1 + ⎢ JV
⎜
⎟⎥ ⎬ ⋅
2 ⎣
2 ⎣
J ∂ξ J ⎠ ⎦ ⎭⎪ ⎩⎪
J ∂η J ⎠ ⎦ ⎭⎪
∂ξ J
∂ξ ⎝ J ∂ξ J
∂η J
∂η ⎝ J ∂η J
⎩⎪
∂ ⎛ h12 ∂ h12 h22 ∂ h11 ⎞ ⎤ ⎪⎫
⎪⎧ Δt ⎡
*( m )
* m −1
( m) ∂ 1
+ν
−
= RHS3 ( )
⎨1 + ⎢ JW
⎜
⎟ ⎥ ⎬ ΔJW
ζ
ζ
ζ
ζ
2
J
J
J
J
J
∂
∂
∂
∂
⎝
⎠ ⎦ ⎭⎪
⎣
⎩⎪
(7)
⎛ Δt
Δt
Δt
*( m −1)
(m) ∂ 1 ⎞ ⎛
( m) ∂ 1 ⎞ ⎛
( m) ∂ 1 ⎞
(m)
⎜1 + JU
⎟ ⋅ ⎜ 1 + JV
⎟ ⋅ ⎜1 + JW
⎟ ΔJk = RHSk
2
2
2
∂ξ J ⎠ ⎝
∂η J ⎠ ⎝
∂ζ J ⎠
⎝
(8)
⎛ Δt
Δt
Δt
*( m −1)
(m) ∂ 1 ⎞ ⎛
( m) ∂ 1 ⎞ ⎛
( m) ∂ 1 ⎞
(m)
⎜1 + JU
⎟ ⋅ ⎜ 1 + JV
⎟ ⋅ ⎜ 1 + JW
⎟ ΔJ ε = RHSε
2
J
2
J
2
J
ξ
η
ζ
∂
∂
∂
⎝
⎠ ⎝
⎠ ⎝
⎠
(9)
∂
∂ξl
m
⎛
∂φ ( ) ⎞ 2 ∂
∗ m
JU l ( )
⎜⎜ Jg li
⎟⎟ =
ξ
ξ
∂
Δ
∂
t
i
l
⎝
⎠
(
)
(10)
where if l = 1, 2,3 , RHSl*( 0) = −Δt ( Cl + Dl + Pl + Al + Rl ) , RHSl*( m) = RHSl*( m −1) + ΔRHSl*( m ) ( m ≥ 1)
n
(
)
(
)
1
ΔRHSl*( m ) = − JU l( m ) − JU l( m −1) − Δt Fl ( m ) − Fl ( m −1) , Fl = Cl + Dl + Pl + Rl
2
Cl =
∂ ( JU iU l )
∂ξi
∂ ( h jk Z k )
⎡
∂ ( ∇ξ l ) ⎤
∂p
∂ ⎡
, Al = −u ⋅ ⎢ JU i
, Pl = Jgli
, Rl = − J α ijkl
⎢ν T
⎥ , Dl = νε lij
∂ξi ⎦
∂ξi
∂ξi
∂ξ k ⎢⎣
⎣
⎛ ∂u ∂u j ⎞ ⎤
⎜⎜ i +
⎟⎟ ⎥
⎝ ∂x j ∂xi ⎠ ⎦⎥
If l = k , ε , RHSl*( 0) = −Δt ( Cl + Dl + Gl ) , RHSl*( m) = RHSl*( m −1) + ΔRHSl*( m ) ( m ≥ 1)
n
(
)
(
)
1
ΔRHSl*( m ) = − Jl ( m ) − Jl ( m −1) − Δt Fl ( m ) − Fl ( m −1) , Fl = Cl + Dl + Gl
2
Cl =
∂
∂
JU i l , Dl = −
∂ξi
∂ξi
⎡⎛
⎛
νT ⎞
ε
ε2 ⎞
∂l ⎤
⎢⎜ν + ⎟ Jgij
⎥ , Gk = − J ( P − ε ) , Gε = − J ⎜ Cε 1 P − Cε 2 ⎟
σl ⎠
k
k ⎠
∂ξ j ⎥⎦
⎝
⎣⎢⎝
The primary difference of implicit SMAC scheme for unsteady and steady incompressible turbulent flows
is the m times Newton iterations between n and n+1 time step for variables. The super-script (m) denotes
the m-th approximation of n+1 time step variables. The computational procedure at each time step for the
advancing solution with a time increment Δt is as follows
1) Compute ΔJl *( m ) ( l = 1, 2,3, k , ε ) from equation (5-9).
2) Compute JU l*( m ) ( l = 1, 2,3) , JU l*( m) = JU l( m −1) + ΔJU l*( m ) .
3) Compute φ (
m)
from equation (10).
1
2
4) Compute JU l( m ) = JU l∗( m) − ΔtJgli
∂φ ( )
( l = 1, 2,3) , Jl ( m ) = Jl ( m −1) + ΔJl ( m ) ( l = k , ε ) , p( m) = p( m −1) + φ ( m) .
∂ξi
m
The procedure is repeated at each time step until get the convergent result.
DISCRETIZATION AND ALGORITHM
The governing equations of implicit SMAC scheme for unsteady incompressible flows were discretized in
MAC staggered grid as shown in Fig. 1.
The momentum equations (5), (6), (7), k equation (8) and ε equation (9) can be solved in the similar
process by dividing into three steps, and applying the first-order upstream-difference to the convective
terms in the left sides. The accuracy of the ΔJl *( m ) ( l = U ,V ,W , k , ε ) depends mainly on the numerical
approximation of the RHSl*( m −1) , and RHSl*( m −1) is affected mostly with the convective term RHSlC*( m −1) . The
higher-order high resolution Chakravarthy-Osher TVD scheme [7] is introduced to discretize the RHSlC*( m −1) .
For example, l = U , in the general curvilinear coordinates, Δξ = Δη = Δζ = h , then
⎯ 308 ⎯
ζ
JU , Jg11 , Jg12 , Jg13
Z 3 , h31 , h32 , h33
JV , Jg 21 , Jg 22 , Jg 23
p, φ , k , ε
u, ς , x , J
Z1 , h11 , h12 , h13
η
ξ
Z 2 , h21 , h22 , h23
JW , Jg31 , Jg 32 , Jg 33
Figure 1: Three-Dimensional staggered grid and definition points of variables
( JU )
2
( RHSC
)
where
( JU )
*( m −1)
1C
=
( m −1)
i , j , kP
h
i , j , kU
2
(
− JU 2
( m −1)
i , j , kP
)
( m −1)
( JUV )i(, j +1,)kZ − ( JUV )(i, j ,kZ)
m −1
i −1, j , kP
+
m −1
3
h
(
( JUW )i(, j ,k +) 1Z − ( JUW )(i, j ,kZ)
m −1
3
+
) (
m −1
2
2
h
)
1
1
= U i(,mj ,−kP1) JU i(,mj ,−k1) + JU i(+m1,−1j ,)k − Dfi ++1/ 2, j , k − Dfi +−1/ 2, j , k −
2
2
1−φ
1+φ
min mod ⎡⎣ Dfi +− 3 / 2, j , k , bDfi +−1/ 2, j , k ⎤⎦ −
min mod ⎡⎣ Dfi +−1/ 2, j , k , bDfi +− 3 / 2, j , k ⎤⎦ +
4
4
1+ φ
1−φ
min mod ⎡⎣ Dfi ++1/ 2, j , k , bDfi −+1/ 2, j , k ⎤⎦ +
min mod ⎡⎣ Dfi −+1/ 2, j , k , bDfi ++1/ 2, j , k ⎤⎦
4
4
( JUV )(i , j ,kZ)
m −1
3
(
) (
)
1 m −1
1
= Vi ,( j , kZ)3 JU i(,mj −−1,1)k + JU i(,mj ,−k1) − Dfi ,+j −1/ 2, k − Dfi ,−j −1/ 2, k −
2
2
1−φ
1+ φ
min mod ⎡⎣ Dfi ,−j +1/ 2, k , bDfi ,−j −1/ 2, k ⎤⎦ −
min mod ⎡⎣ Dfi ,−j −1/ 2, k , bDfi ,−j +1/ 2, k ⎤⎦ +
4
4
1+ φ
1−φ
min mod ⎡⎣ Dfi ,+j −1/ 2, k , bDfi ,+j − 3 / 2, k ⎤⎦ +
min mod ⎡⎣ Dfi ,+j −3 / 2, k , bDfi ,+j −1/ 2, k ⎤⎦
4
4
( JUW )i(, j , kZ)
m −1
2
(
) (
)
1 m −1
1
= Wi ,( j , kZ)2 JU i(,mj ,−k1) + JU i(,mj ,−k1−)1 − Dfi ,+j , k −1/ 2 − Dfi ,−j , k −1/ 2 −
2
2
1−φ
1+ φ
min mod ⎡⎣ Dfi ,−j , k +1/ 2 , bDfi ,−j , k −1/ 2 ⎤⎦ −
min mod ⎡⎣ Dfi ,−j , k −1/ 2 , bDfi ,−j , k +1/ 2 ⎤⎦ +
4
4
1+ φ
1−φ
min mod ⎡⎣ Dfi ,+j , k −1/ 2 , bDfi ,+j , k − 3 / 2 ⎤⎦ +
min mod ⎡⎣ Dfi ,+j , k − 3/ 2 , bDfi ,+j , k −1/ 2 ⎤⎦
4
4
(
= (V
= (W
Df i +±1/ 2, j , k = U i(, j , kP) ± U i(, j , kP)
Df i ,±j −1/ 2, k
Df i ,±j , k −1/ 2
m −1
m −1
( m −1)
i , j , kZ 3
± Vi (, j , kZ)3
( m −1)
i , j , kZ 2
m −1
) ( JU
) ( JU
) ( JU
± Wi ,( j , kZ)2
m −1
( m −1)
i +1, j , k
− JU i(, j , k )
m −1
)
2
( m −1)
i, j ,k
− JU i(, j −1,)k
)
2
( m −1)
i, j ,k
m −1
− JU i(, j , k −)1
m −1
)
2
The flux limiter function is defined as min mod ( x, y ) = sign ( x ) max ⎡⎣0, min ( x , ysign ( x ) ) ⎤⎦ . The subscripts U, V,
W, P and X show the points where JU, JV, JW, p and x are defined respectively. The Chakravarthy-Osher
TVD scheme is third-order upwind difference while φ = 1 3 and b = 4 .
The discretized linear equations of equations (5-9) at each step have a diagonally dominant tridiagonal
matrix, and can be computed easily by CTDMA method. The Poisson equation (10) is discretized by
second-order central-difference scheme. The corresponding linear equations are computed by
Tschebyscheff SLOR (Successive Linear Over Relaxation) method, and accelerated by ADI method.
⎯ 309 ⎯
APPLICATION TO UNSTEADY CASCADE FLOWS
A numerical simulation of unsteady 2-D cascade flow was performed using the present implicit SMAC
scheme. The blade section in cascade with a pitch-chord ratio 1.0 is NACA 65-410. The settled angle of
cascade was 30o. The H-shaped grid was generated algebraically as shown in Fig. 2. Where the grid was
concentrated near the blade surface and the trail of the blade in order to simulate the boundary layer and
the wake flows accurately.
Figure 2: Computational grid (151×75)
The periodic condition was imposed on the cyclic boundaries in the upstream and the downstream regions.
The Neumann condition for the pressure and the no-slip condition were also given on the blade boundaries.
The uniform velocity profile was given on the inlet boundary and p = 0 on the outlet boundary. The k
and ε were given directly by the velocity profile on the inlet boundary. On the blade boundary, the k
and ε were determined by wall function. All of the other boundaries were given Neumann condition of
unknown variables.
Reynolds number was defined as Re = u ⋅ C ν , u was uniform velocity magnitude on the inlet boundary.
C was chord length. Fig. 3 shows the numerical results for the angle of attack α = 90 and Re = 103 by
using 121×55 grid points. In this Reynolds number, the boundary layer and wake flows were simulated
well. There were no separation both on suction side and pressure side of the blade.
1
Cp
0.5
0
0
0.5
1
X/C
-0.5
-1
Figure 3: Steady flow at α = 90 and Re = 103
1
Cp
0.5
0
0
0.5
1
X/C
-0.5
-1
Figure 4: Instantaneous cascade flow at α = 90 and Re = 104
Fig. 4 shows the velocity vector, pressure contours and the surface pressure distribution of the cascade
flow for the angle of attack α = 90 and Re = 104 by using 151×75 grid points. The flow becomes unsteady
as indicated in Fig. 4 which shows the detailed flow aspect near the trailing edge. The Kármán Vortex
Street is formed in the downstream of cascade. The separation point is at the suction side of the blade,
where is two-third chord length to the head of the blade. In Fig. 5, the unsteady cascade flow is shown at
different time. As a result, the fluctuating flows is periodic. The period is about 0.5.
⎯ 310 ⎯
1
1
t=0.00
0
0 .5
Cp
Cp
0
t=0.08
0.5
0 .5
1
0
0
0.5
-0 .5
-0.5
-1
-1
1
1
t=0.16
t=0.24
0 .5
0
0.5
Cp
Cp
0.5
0
1
0
0
0.5
X /C
1
X /C
-0 .5
-0.5
-1
1
-1
1
t=0.32
0
0
t=0.40
0.5
0 .5
Cp
0.5
Cp
1
X /C
X /C
1
0
0
0.5
X /C
1
X /C
-0.5
-0.5
-1
-1
Figure 5: Unsteady cascade flow at α = 90 and Re = 104
Fig. 6 shows the fields of the cascade flow for the angle of attack α = 90 and Re = 2.45 × 105 by using
151×75 grid points. The transformation of the vortex on the surface of the blade is seen clearly.
t = 0.00
1
Vortex C
Vortex C
Vortex A
0.5
Vortex A
Vortex A
t = 0.13
Vortex A'
0.5
Vortex A'
Cp
Cp
Vortex A
1
Vortex C
Vortex C
Vortex B
0
Vortex B
0
0.5
Vortex B
1
X/C
0
Vortex B
0
-0.5
1
t = 0.26
Vortex A
Vortex A'
t = 0.39
Vortex A Vortex A' Vortex A''
0.5
Vortex A Vortex A' Vortex A''
Vortex B
Vortex B'
0.5
Cp
Cp
Vortex A'
0
Vortex B
0
0.5
Vortex B
1
X/C
Vortex B'
Vortex B
Vortex B'
0
1
Vortex A' Vortex A''
t = 0.76
0.5
Vortex A
Vortex A'
Vortex A
Vortex A'
0.5
Cp
Cp
Vortex B
Vortex B
0
0
0.5
0
1
X/C
Vortex B'
Vortex B'
1
t = 0.89
Vortex A'
Vortex A
Vortex B'
Vortex A'
Vortex A
t = 1.02
Vortex A'
0.5
0.5
0
0.5
Vortex A''
Cp
Vortex A''
0
0
1
X/C
Vortex B'
1
-0.5
Cp
Vortex A
0.5
X/C
1
Vortex A'
0
Vortex B'
-0.5
Vortex A
1
-0.5
t = 0.51
Vortex A
Vortex B'
0.5
X/C
1
Vortex A' Vortex A''
0
Vortex B'
-0.5
Vortex A
1
-0.5
1
Vortex A
0.5
X/C
0
0.5
1
X/C
Vortex B'
-0.5
Vortex B'
-0.5
Figure 6: Unsteady cascade flow at α = 90 and Re = 2.45 ×105
To validate the result, in Fig. 7, the time-averaged surface pressure distribution at the angle of attack
α = 90 and Re = 2.45 × 105 derived from simulation and experiment are compared. The result simulated using
⎯ 311 ⎯
the fine grid 255×75 is more accurate than one using the coarse grid 151×75 to the experimental data.
However, at the head and the trail surface of the blade, the numerical results derived both coarse grid and
fine grid differ somewhat obviously from the experimental data. The possible reason is the H-shaped grid
which has not good fitness to the blade. To investigate the unsteady flow characters of the cascade, the
flow fields at different angle of attack are also computed.
1
________
255X75
________
151X75
o
EXPERIMENT
Cp
0.5
0
0
0.5
1
X/C
-0.5
-1
Figure 7: Time-averaged surface pressure distribution at α = 90 and Re = 2.45 × 105
CONCLUSIONS
1) The present implicit SMAC scheme can be applicable to simulate the unsteady incompressible turbulent
flows.
2) The unsteady flow of the cascade is periodic. There is flow separation on the suction side of the blade.
The Kármán Vortex Street is formed in the downstream of the cascade.
3) The Reynolds number and the angle of attack are two primary values which affect the characteristics of
the unsteady cascade flow.
4) The computational results of surface pressure distribution were in good agreement with the experimental
data. The quality of the grid is one of the important factors for the accuracy of the numerical simulation.
REFERENCES
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3. Rosenfeld M, Kwak D, Vinokur M. A fractional step solution method for the unsteady incompressible
Navier-Stokes equations in generalized coordinates systems. J. Comput. Phys., 1991; 94(1): 102-113.
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6. Zhang Y, Zhao Y, Cao S et al. Implicit SMAC scheme for the incompressible Navier-Stokes
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