Journal of applied science in the thermodynamics and fluid mechanics
Vol. 1, No. 1/2007, ISSN 1802-9388
NUMERICAL SOLUTION OF COMPRESSIBLE STEADY FLOWS
IN A 2D CHANNEL
*Pavel KRYŠTŮFEK, **Karel KOZEL
*Technical University of Liberec, Department of Power Engineering Equipment
Studenstká 2, 46117, Liberec 1, Czech Republic
Phone: +420485353438, Fax: +420485353644, E-mail: pavel.krystufek@tul.cz
**Technical University in Prague, Faculty of Mechanical Engineering
Department of Technical Mathematics
Karlovo náměstí 13, 121 35, Prague, Czech Republic
Phone: +420224357301, Fax: +420224911406 Email: kozelk@fsik.cvut.cz
The work deals with numerical solution of the system of Euler equations describing steady two dimensional
inviscid compressible fluid flows in 2D channel and the system of Navier-Stokes equations describing steady
two dimensional laminar viscous compressible fluid flows in a channel.
Keywords: numerical, solution, compressible, steady, flows, channel
1 INTRODUCTION
Numerical solution of the flow problems has come through continuous development in last years. More often
this method has exploited in commercial sphere. The companies use it for cost reduction of development of
new products. It economizes the costs of experimental measurement, of model creation and mainly steals
the time needed for the prototype development.
We are able to find analytical solutions of equations describing flows for few events, namely for only extra
simple situations. Therefore we understand to the creation of the theoretic solution of the physical model, in
consequence its mathematical analogy – the numerical approximation and the numerical solution. We divide
the physical models into potential, inviscid and viscous. The mathematical model for inviscid flow is
described by Euler’s equations. A viscous fluid flow is described by Navier-Stokes equations. The viscous
model can be laminar or turbulent.
2 PROBLEM
Use D We want to solve flow in a GAMM channel, which is presented in this work. In the following part (in the
future) the solution will be extended for the flow over the profiles NACA 0012, RAE 2822 and on grid SE
1050. Solving the inviscid flow we have used system of Euler’s equations.
3 MATHEMATICAL MODEL
Wt  F W  X  G W Y  R WX ,WY  X  S WX ,WY Y
(1)
Equation describes the system of Navier-Stokes equations in conservative form for unsteady flow.
F W  X  G W Y  R WX ,WY  X  S WX ,WY Y
(2)
The system we obtain by solving unsteady form (1) with steady boundary conditions and with t   . This
method is time dependent method.
This system is completed by state equation for the ideal gas.
 1

p    1 e    w12  w22 
2


The conservative variable
(3)
W is defined
W  
 w1
1
 w2
e
T
(4)
Journal of applied science in the thermodynamics and fluid mechanics
Vol. 1, No. 1/2007, ISSN 1802-9388
The function of inviscid physical fluxes F , G and the function of viscous physical fluxes R , S are then
defined
F W     w1
 w12  p  w1w2
 e  p  w1  ,
G W     w2
T
 w1w2
 w22  p
 e  p  w2  ,
T
(5)
T

R WX ,WY    0  11  12

k 
w1 11  w2 12  T  ,
 

S WX ,WY    0  21  22

k 
w1 21  w2 22  T  ,
 
T
(6)
where  is density, ( w1 , w2 ) is velocity, p is pressure and e is total energy per unit volume,  is tensor of
shear tension, T is temperature.
Dynamic viscosity depends on temperature and can be calculating it for example from Sutherland function
(7). In function are occur: 0 reference dynamic viscosity for T0 , T0 – reference temperature, S – effective
temperature.
4 CONSIDERED MESH OF THE FINITE VOLUME METHOD
We selected for numerical solution a structured mesh formed by quadrilateral finite volumes. Figure 1 shows
mesh with part of circular arc with selecting parameters. A program for generating mesh makes it possible
any thickening of mesh in x direction. In y direction is controlled density only one parameter.
5 NUMERICAL SOLUTION
We obtain numerical solution by finite volume method for a case of structured mesh of quadrilateral cells. We
integrate Navier-Stokes equations over computation cell (Figure 2) and we apply Green theorem, we get
expression (7)
 W
Di , j
t2
t1
dxdy 
  Fdy  Gdx     Rdy  Sdx  .
Di , j
Di , j
Applying mean value theorem, we obtain form (8), where are
Wi ,nj1  Wi ,nj 
t
i , j
i , j 
 computation cell area, t time step (9)
   F  R  dy  G  S  dx  .
Di , j
 dxdy ,
t  t2  t1
Di , j
Figure 1: Structured mesh formed tetragonal elements
2
(7)
(8)
(9)
Journal of applied science in the thermodynamics and fluid mechanics
Vol. 1, No. 1/2007, ISSN 1802-9388
Di,j+1
j
2
A3
Di,j
Di-1,j
i
A2
1
Di+1,j
3
A1 = A5
4
A4
Di,j-1
Ak =[xk ,yk ]
Figure 2: Computation cell
5.1 Lax-Friedrich finite volume method
Form (10) describes Lax-Friedrich scheme for Euler equations. Parameter    0,1
. Lax-Friedrich scheme is
first-order accuracy
Wi ,nj1  Wi ,nj 
N
t
 N
n
n
F

y

G

x



Wkn  Wi,nj  .
 k k k k N
  Di , j  k 1
k 1
(10)
We can rewrite the form (11) for structured mesh:
Wi ,nj1  Wi ,nj 
t
i , j
F
4
k 1
Fkn 
n
k
yk  Gkn xk  

W
4
n
i 1, j
1
 F Wi ,nj   F Wkn   ,

2
 2Wi ,nj  Wi n1, j  
Gkn 
yk  yk 1  yk ,

4
W
n
i , j 1
 2Wi ,nj  Wi ,nj 1  ,
1
G Wi ,nj   G Wkn   ,

2
xk  xk 1  xk .
(11)
(12)
(13)
5.2 Lax-Wendroff finite volume method
Forms (14, 15) describes Lax-Wendroff scheme for Euler equations in case of Richtmyer’s modification. This
scheme is already second-order accuracy. The component AD Wi ,nj  is artificial dissipation of second-order.
Wi ,nj1  Wi ,nj 
N
t
 N
Fkn yk  Gkn xk    Wkn  Wi ,nj 


N k 1
  Di , j  k 1
(14)
2t N
  Fkn 1yk  Gkn 1xk   AD Wi ,nj 
  Di , j  k 1
(15)
Wi ,nj 2  Wi ,nj 
For McCormack’s modification of Lax-Wendroff scheme we obtain form (18).
Wi ,nj1/ 2  Wi ,nj 
1
t
Wi ,nj1  Wi ,nj  Wi ,nj1/ 2 

2
i , j
t
i , j
F
4
k 1
 F
4
k 1
n 1/ 2
k
3
n
k
yk  Gkn xk 

yk  Gkn 1/ 2 xk    AD Wi ,nj 


(16)
(17)
Journal of applied science in the thermodynamics and fluid mechanics
Vol. 1, No. 1/2007, ISSN 1802-9388
The physical fluxes are then:
F1n 1/ 2  F2n 1/ 2  F Wi ,nj1/ 2 
F1n  F Wi n1, j 
, F3n 1/ 2  F Wi n1,1/j 2 
F2n  F Wi ,nj 1 
F  F  F W
n
3
n
4
n
i, j

n 1/ 2
4
F
 F W
n 1/ 2
i , j 1
.
(18)

5.3 Artifical dissipation
The artificial dissipation damps undesirable oscillations. We used relation:
AD Wi ,nj   k1 i Wi n1, j  2Wi ,nj  Wi n1, j   k 2 j Wi ,nj 1  2Wi ,nj  Wi ,nj 1 
i 
pin1, j  2 pin, j  pin1, j
pin1, j  2 pin, j  pin1, j
, j 
pin, j 1  2 pin, j  pin, j 1
pin, j 1  2 pin, j  pin, j 1
(19)
(20)
The coefficients k1 , k2 aren’t resolutely set. We should be found them comparing solutions with results of
experiments. In the work we used values 0.2, 0.8, 1, 2. We used for artificial dissipation of first-order values
1/10, 1/5 and 1.
5.4 Stability
Choice of time step is given by stability requirements. For Navier-Stokes equations:
t 
K .CFL
,
ws  a wn  a
1 
 1

 2  2  2 
s
n
n 
 s
(21)
K .CFL
,
ws  a wn  a

s
n
(22)
for Euler’s equations:
t 
The constant K is set from numerical experiments,
ws is velocity in tangent direction, wn is velocity in
normal direction and CFL is Courant number.
5.5 Convergence
We followed convergence by residual behaviour, residual is computed by relation (25).
1
Re zW 
m.n
n
i, j
 Wi ,nj1  Wi ,nj


t
i, j 
2

 ,

v L2   .
(23)
The residual lead into graph in logarithmic scale depends on time (iterations number).
6 BOUNDARY AND INITIAL CONDITIONS
On inlet are set values approaching flow. On outlet was used extrapolation. On wall was used condition of
zero derivation of velocity vector along normal – reflection method (24).
w
0
n
4
(24)
Journal of applied science in the thermodynamics and fluid mechanics
Vol. 1, No. 1/2007, ISSN 1802-9388
The initial conditions must agree with request of even input approaching flow. That was defined by Mach
number Ma , size of density and absolute size of velocity, where  is an angle between centers
neighbouring cells in x direction (25).
1


 n1  
Ma
.cos

  
n  
W   2    Ma.sin    .
 n3  

1
   1
2
 n4      1  2 Ma 


(25)
7 RESULTS
Numerical solution has been applied on structured mesh with 40x80 and 60x120 cells. Effect of artificial
dissipation and effect of mesh density was tested in Lax-Friedrich scheme. Effects of parameters of artificial
dissipation k1 , k2 , and effect of mesh density was tested in modifications of Lax-Wendroff scheme.
Calculations witch upping Mach number was performed for McCormack modification. On next results can be
seen creation of shock wave witch upping Mach number. An obtained result was compared witch results in
publication [1].
We found, that the results with strong artificial dissipation AD W  are smoothed. Zierep’s singularity
disappeared. Results with low artificial dissipation AD W  oscillate near discontinuities. According to
personal numerical experiments is optimal value of k1 , k2 between 0.8 and 1.
Ma
  0.1
Ma na stěnách
2
1.4
Spodní stěna
Horní stěna
1.2
1.3
1.1
1.5
1.2
1
1.1
1
1
Ma
y
0.9
0.5
0.9
0.8
0.8
0.7
0
0.7
0.6
0.6
-0.5
0.5
0.5
-1
-1.5
-1
-0.5
0
x
0.5
1
0.4
-1.5
1.5
-1
-0.5
0
x
0.5
1
1.5
Figure 3: With diminishing value  is effect of artificial dissipation smaller and thereby also punctuate shock wave on
bottom wall (increases also Mach number) - L-F, fine mesh, K=0.8.
 1
Ma na stěnách
Ma
2
Spodní stěna
Horní stěna
1.6
1.5
k1  k2  0.2
1.5
1.4
1.5
1.4
1.3
1.3
1.2
1
1.2
1
0.5
0.9
1.1
Ma
y
1.1
1
0.9
0.8
0
0.8
0.7
0.7
0.6
-0.5
0.6
0.5
0.5
-1
-1.5
-1
-0.5
0
x
0.5
1
0.4
1.5
-1.5
-1
-0.5
0
x
0.5
1
1.5
Figure 4: Zierep’s singularity is distinct. Artificial dissipation AD W  is added and her size was controlled by
k1 , k2 - L-W-R, fine mesh, K=0.4
5
Journal of applied science in the thermodynamics and fluid mechanics
Vol. 1, No. 1/2007, ISSN 1802-9388
M  0.675
Ma na stěnách
Ma
1.5
2
Spodní stěna
Horní stěna
1.3
1.4
1.5
1.3
1.2
1.2
1.1
1.1
1
1
0.9
Ma
y
1
0.5
0.9
0.8
0.8
0.7
0
0.7
0.6
0.5
-0.5
0.6
0.4
0.5
-1
-1.5
-1
-0.5
0
x
0.5
1
-1.5
1.5
-1
-0.5
0
x
0.5
1
1.5
Figure 5: We see increasing Mach number on profile. For input Mach number 0.675 has arisen already shock wave.
Zierep’s singularity is visible - L-W-MC, fine mesh, K=0.4,
k1  k2  1 .
8 COMPARISON
If we compare our results with results of the other authors, we can declare, that the results are in good
agreement. Achieved results confirmed the properties of used schemes and it succeed determine acceptable
parameters for artificial dissipation  and k1 , k2 . In L-F scheme was validated strong effect of artificial
dissipation on „smoothing“ of numerical solution and necessity to create softer mesh for increasing accuracy.
Using second-order scheme very good and accuracy results have been achieved. Similar results have been
achieved by suitable choice of k1 , k2 (artificial dissipation parameter).
Ma na stěnách
1.5
Spodní stěna
Horní stěna
1.4
1.3
1.2
1.1
Ma
1
0.9
0.8
0.7
0.6
0.5
0.4
-1.5
-1
-0.5
0
x
0.5
1
1.5
Figure 6: Results comparison
REFERENCES
[1] KRYŠTŮFEK, P. – KOZEL, K.: Numerical solution of compressible steady flows in a channel, Technical
University of Liberec, Liberec 2004
[2] DVOŘÁK, R. – FURST, J.: Numerické metody řešení problému proudění I, ČVUT, Praha 2001
[3] TESAŘ, V.: Mezní vrstvy a turbolence, ČVUT, Praha 1996
[4] MARYŠKA, J. – ŠEMBERA, J.: Mechanika tekutin, Technical University of Liberec, Liberec 2002
[5] DVOŘÁK, R. – FURST, J.: Numerické metody řešení problému proudění I, ČVUT, Praha 2001
[6] BRDIČKA, M. – SAMEK, L. – SOPKO, B.: Mechanika kontinua, Academia Praha, 2000
[7] FEISTAUER, M.: Mathematical Methods in Fluid Dynamics, Longman Harlow 1993
[8] WHITE, F., M.: Fluid Mechanics – Fourth Edition, McGraw-Hill, 2001
[9] FOŘT, J. – JIRÁSEK, A. – KOZEL, K.: Numerical solution of the inviscid flow over a profile, Letecký
zpravodaj 1/1998
6