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```					Chapter Five                                  Two Dimensional Nozzle Flow

Chapter Five
Two Dimensional Nozzle Flow

5.1    Introduction.

In the present chapter the governing equations of Euler two
dimensional flow of supersonic flow are solved. Equations to be used in the

numerical schemes are presented; starting with the general form, advancing
to the vector form of unsteady Euler equation.
The important concepts in solving the governing equations using
finite difference method are discussed in this chapter. First, the governing
equations are written in conservative form. Thereafter, the finite difference
form of the equation are presented, together with the initialization used for
simultaneous equations and details of the implementation of boundary
conditions are discussed. Inherently, the governing equations are an
extremely stiff, non–linear system of equations and therefore their
numerical solution deserves special care–finite difference method converts
partial differential equations into a system of algebraic equations. These
equations are then written at the discretized grid points used to approximate
the flow domain. The system of equations then is solved explicitly, where
finite difference form of the differential equation is written so that only one
unknown appears in the equation.
The flow analysis model used in the present work is based on time
marching of time dependent Euler equations.

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Chapter Five                                                        Two Dimensional Nozzle Flow

5.2    Governing Equations.
The mathematical behavior of the Euler partial differential equation
is classified as elliptic in subsonic flow and hyperbolic in supersonic flow.
If the time dependent terms are retained and a steady state is assumed as in
current work the solution can be obtained using time marching procedure,
by marching from some initial guessed flow field through time until a

steady state is obtained, where all domain expressed in hyperbolic
differential equation.
The governing equations for an inviscid, compressible, two
dimensional flow expressed in a conservative form are [9]:
Continuity equation:

 ρ  ρu   ρ v 
 .ρ v                                  .......... .......... .......... .......... .......... .......... .......... ..(5.1 a)
t    x      y

The conservation of momentum equation is;

 ρu                
t

x

ρ u2  P 
y
ρ u v   0.0                 .......... .......... .......... .......... .......... .......... (5.1 b)

 ρv  
    ρ u v    ρ v 2  P  0.0
                         .......... .......... .......... .......... .......... .......... ..(5.1 c)
t     x            y

The conservation of energy equation is;

 e t  
    e t  P  u   e t  P  v   0.0           .......... .......... .......... .......... .......... .....(5.1 d)
t      x                  y

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Chapter Five                                                        Two Dimensional Nozzle Flow

Where , u, v, P, et are the density, axial velocity, radial velocity, pressure,
and internal energy, respectively.
It is convenient to combine the governing equations, into a compact
vector before applying a finite difference algorithm to these equations. The
Euler equations (5.1a through 5.1d) in Cartesian coordinates may be
written in a vector form as:

U  E  F
       0.0            .......... .......... .......... .......... .......... .......... .......... .......... ......... (5.2)
t   x y

Where:

ρ            ρ u                                     ρ v            
ρ u           2                                      ρ v u          
U        , E  ρu  P               , and           F                
ρ v          ρ u v                                   ρ v 2  P 
                                                                    
ρ e t        ρ e t  P  u
                                        ρ e t  P  v 
               

5.3 Discretization technique of Euler equations.

After writing the Euler governing equation in vector form the key
steps in applying MacCormack’s technique are shown in the following
subsection.

5.3.1      Explicit MacCormack’s scheme.
Explicit time dependent solution of the two dimensional Euler

equations has been performed using MacCormack’s predictor–corrector
finite difference technique.

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Chapter Five                                                               Two Dimensional Nozzle Flow

In chapter three, the MacCormack’s time marching technique in one
spatial direction is applied; down the length of convergent– divergent
nozzle. In the present chapter the MacCormack’s time– marching technique
will used to march down–stream, effectively solving for flow properties in
two spatial dimensions in the convergent–divergent nozzle problem with
march in time to steady state solution by solving the flow properties at

every (i, j) spatial location. Equation (5.2) is rewritten as:

U    E F
                      .......... .......... .......... .......... .......... .......... .......... .......... .......... .....(5.3)
t    x y

Due to advantages discussed previously in chapter four, equation (5.2)
is transformed to a computational domain where grid points spacing are
uniform and the domain is rectangular. The transformed Euler equation
(5.2) in curvilinear coordinates written in vector form can be given in
Appendix [A].

U  E  F
         0.0                  .......... .......... .......... .......... .......... .......... .......... .......... ........ (5.4)
t   ξ   η

Where U , E and F are vector given by:

U
U                    ..........
..........              ..........
..........      ..........
..........
..........      ..........
..........              .......... (5.5)
..........      ....
J

E
1
J

ξ x E ξ y F               ..........
..........      ..........      ..........
..........      ..........      ..........
..........              ......
.......... (5.6)

F
1
J

η x E η y F                ..........
..........      ..........      ..........
..........      ..........      ..........
..........              .....
.......... (5.7)

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Chapter Five                                                                     Two Dimensional Nozzle Flow

By means of a Taylor series expansions, the flow– field variables are
advanced at each grid point (i , j) in step of time, as shown below:

U
U i,Δt  U i, j  (
t         t
)av. Δt        .......... .......... .......... .......... .......... .......... .......... .......... ....(5.8)
t
j

Where, once again, U is a flow field variable assumed known at time t,

either from initial condition or as a result from the previous iteration in
 U 
time. 
            av. is defined as:

 t 

1   U        U  
t           t Δt
 U 

  t   2   t     t  
                                                   ..........      ..........
..........      ..........      ..........
..........              (5.9)
..........
      av.   
       i, j      i, j 

 U 
To obtain a value of 
                          av. (above) so that the solution can be advanced,

 t 

the following steps are taken:

t
 U 
1):            t  is a calculated using forward spatial difference on the right
     
      i, j

hand side of the governing equations from the known flow field at
time t.

2): From step 1, predicted values of the flow– field variables (denoted by a
bar ) can b obtained at time t  Δt , as follows:

t
 U 
U    t Δt
i, j     U    t
i, j   
  t  Δt

.......... .......... .......... .......... .......... .......... .......... .......... ....(5.10)
      i, j

Combining steps 1 and 2, predicted values are determined as follows:

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Chapter Five                                                                         Two Dimensional Nozzle Flow

U i,t Δt  U i, j 
j
t       Δt
Δξ

E it 1, j  E i, j 
t     Δt
Δη
             
Fi,t j  1  Fi, j                   .......... .......... .......... .......... ...... (5.11)

3): Using reward spatial differences, the predicted values ( From step 2 ) are
inserted into the governing equations such that a predicted time
t Δt
 U              
derivative 
 t              
         can be obtained.
                  i, j
t Δt
 U                       
4):     Finally, substitute 
 t                       
         (from step 3) into Equation (5.9) to obtain
                           i, j

corrected second order accurate values of U at time t Δt . As in

Equation (5.11) steps 3 and 4 are combined as follows:

U i,Δt 
t
j
1 t
2
t Δt Δt    t Δt

t Δt    Δt    t Δt

t Δt 
U i, j  U i, j  Δξ E i, j  E i 1, j  Δξ Fi, j  Fi, j 1                                     ........... .......... (5.12)


Steps 1 to 4 are repeated until the flow– field variables approach a steady
state value; this is the desired steady state solution.

After each predictor or corrector step, the primitive variables are
obtained by decoding the U vector, as shown below;

ρ  U1          .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ...(5.13 a)

ρ u U2
u                         .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ..(5.13 b)
ρ   U1

ρ v U3
v                         .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ..(5.13 c)
ρ   U1

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Chapter Five                                               Two Dimensional Nozzle Flow

     v2 
ρ  et    U 4

     2 

Or
U 4 u2  v 2
et                            ..........
..........        ..........
..........
..........
..........        ..........
..........                 .(5.13d)
..........
U1     2

With ρ , u , v , and e t determined, the remaining flow– field properties can be

obtained by using the equations:

et
T        , P  ρ RT
CV

5.3.2 Calculation of step sizes in time (Stability criterion ).

In general, for explicit methods, the value of t can not be arbitrary,

rather it must be less than some maximum values for stability. The time
dependent applications described in section (5.3.1) deals with governing
flow equations that are hyperbolic with respect to time. Then, it was stated

that t must obey the Courant–Friedrichs–Lowry criterion (CFL). The CFL
criterion states that physically the explicit time step must be no greater than
the time required for a sound wave to propagate from one grid to next.
MacCormack experience recommended that CFL factor should be as closed
to unity as possible but depend upon actual application, the maximum
allowable value of CFL factor for stability in explicitly time dependent

finite difference calculation can vary from approximately 0.5 to 1.0 on the
other hand side, the CFL is a function of fluid velocity and speed of sound
and there is variation with the spatial coordinate, then the local value of t

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Chapter Five                                                 Two Dimensional Nozzle Flow

associated with each grid point will be different from one point to the next.
Finally, the value of time step that subjected to a stability criterion
employed should be minimum overall the grid points.
To determine the value of time step, the following version of the CFL
criterion [22] is used. Where ai,j is the local speed of sound in meters per
second, and C is the courant number.

1
 U i, j Vi, j           1      1     
( Δt )i, j                a i, j              
 Δξ      Δη            Δξ     Δη 2   
CFL
2
                                     

Where:


Δt  min C( Δt )i, j
CFL
           ..........
..........              ..........
..........              ..........
..........
..........                      .......... (5.14)
..........      ....

For      0.5  C  0.8

5.3.3     Initial and boundary conditions.
The description of a system of differential equations is not complete
without the specifications of initial and boundary conditions. Once the

problem has been specified, an appropriate set of governing equations and
boundary conditions must be selected. It is generally accepted that the
conservation of mass, momentum, and energy govern the phenomena of

importance to the field of fluid dynamics. These may be steady or unsteady
and compressible or incompressible boundary types, which may be
encountered, and include solid walls, inflow and out flow boundaries.

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Chapter Five                                                                    Two Dimensional Nozzle Flow

Because the solution is marched from a set of initial conditions, the
flow properties must be specified at each (i, j) location at time t = 0.0.
Having the specified the initial conditions, the equation are marched
in time to the steady state solution. In that process, conditions must be
enforced at the boundary of the computational domain. Figure (5.1)
demonstrated the inflow and outflow boundary conditions.

Computational domain
Reservoir properties

Outflow properties
extrapolated from
P = Po                                                                                     interior points
T = To                                                 u(i-2,j) u(i-1,j) u(i,j)
 = o

Figure (5.1):
Inflow and outflow boundary
conditions

This figure shows that the inflow boundary conditions are the same

for reservoir conditions, and outflow boundary conditions are calculated
based on an extrapolation from the two adjacent interior points at the same j
location. For example, u is determined as follows:

u(N, j)  2 u(N 1, j)  u(N  2, j)          .......... .......... .......... .......... .......... .......... .......... .......... ...(5.15)

While the surface boundary conditions ( Wall of nozzle) is calculated at the
following subsection

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Chapter Five                                                                  Two Dimensional Nozzle Flow

5.3.3.1 Wall boundary conditions.

At grid body surface, tangency must be satisfied for inviscid flow. The
components of the momentum equation for the two– dimension flow may
be expressed as:

                               1
(ρ u)  (ρ u 2  P )  (ρ uv)  (ρ uv)  0.0                                       .......... .......... .......... .......... ..(5.16)
t        x             y       y

                              1
(ρ v)  (ρ uv)  (ρ v 2  P)  (ρ v 2 )  0.0                                   .......... .......... .......... .......... .....(5.17)
t        x       y            y

By definition

m ρ v . n
        ˆ       .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ...... (5.18)

And is equal to zero at the wall of nozzle. Since,

η
n
ˆ
η

ˆ j
v  (u i  vˆ)

And
ˆ                    ˆ
η η x i  η y ˆ  J (  yξ i  xξ ˆ)
j                   j

Substitution of these relations into Equation (5.18) yields:
ρ vxξ  ρ uyξ  0.0            .......... .......... .......... .......... .......... .......... .......... .......... .......... ...(5.19)

At time derivative of Equation (5.19) provides

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Chapter Five                                                Two Dimensional Nozzle Flow

            
xξ      (ρ v)  yξ (ρ u)  0.0       ..........        ..........
..........                 ..........
..........                 .......... (5.20)
..........        .......
t           t

The grid system has been assumed to be independent of time. With some
mathematical manipulation between metric transformations and momentum
equation the following results:

                                                
xξ      (ρ u 2  P)  yξ    (ρ u v)  xξ    (ρ u v)  xξ    (ρ v 2  P)  0.0             .......... ...(5.21)
x                  y              x              y

Equation (5.21) is rearranges and transformation to body fitted coordinate

and is in conservative form. Therefore, the conservative form of Equation
(5.21) is expressed as:

 ρ uU        ρ uU        ρ uV        ρ uV 
ηx        η y        η x        η y       
 J ξ         J ξ         J η         J η                                     .......... (5.22)
..........        ......
ξ P        η P        ξ y P      ξ  P 
η x  x  η x  x  η y      J  η y  J   0.0
           
 J ξ        J η             ξ          η

Where U is the tangential velocity in computational domain.
In order to obtain a finite difference Equation (5.22), a second order

central difference approximation for the  derivatives and a second – order
one sided difference approximation for the  derivatives are used. The
unknowns are the values for pressure at the wall of nozzle. The values at

the interior points have already been computed with the second order
approximation described in subsection (5.3.1), the finite difference
equation is obtained as:

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Chapter Five                                                                Two Dimensional Nozzle Flow

η xi, j  ρ uU    ρ uU       η yi, j                      ρ uU       ρ uU      
                                                                   
2Δ ξ  J  i 1,1  J  i 1,1  2Δ ξ
                                                     J  i 1,1  J  i 1,1 
                          

η xi, j   ρ uV      ρ uV    ρ uV  
            3      4              
2Δ η   J  i,1
              J  i,2  J  i,3 


η yi, j   ρv V     ρ v V   ρ v V                             
            3      4                                         
2Δ η   J  i,1
              J  i,2  J  i,3                           


η yi, j  ξ x P   ξ P         η xi, j   η x P     η P     η P  
                    x                  3       4 x    x  
2Δ ξ  J  i  1,1  J  i 1,1  2Δ η   J  i,1
                                                J  i,2  J  i,3 


η yi, j  ξ y P   ξ y P     
                             
2Δ ξ  J  i 1,1  J  i 1,1 
                   

η yi, j   η y P      η P      η P  
            3       4  y    y    0.0                                                 ..........
..........              .......
.......... (5.23)
2Δ η   J  i,1
        
 J 

    
 i,2  J  i,3 


The value of V is equal to zero at the wall of nozzle, and therefore,
those terms are dropped. This equation is regrouped so that a tridiagonal
system is formed.
The rearrangement is as follows:

a i Pi 1,1  bi Pi,1  c i Pi  1,1  d i        .......... .......... .......... .......... .......... .......... .......... ...... (5.24)

Where:

1             ξ                   ξ y           
ai              η xi,1  x          η yi,1 
 J    
        
2Δ ξ    
        J  i 1,1                 i 1,1 


3 η x η y            
2   2

bi                          
2Δ η  J


 i,1

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Chapter Five                                                             Two Dimensional Nozzle Flow

1          ξ                   ξ y             
ci         η xi,1  x          η yi,1 
 J      
        
2Δ ξ 
        J  i 1,1                   i 1,1 


Pi,3        η            η y         Pi,2                 η            η y    
di             η xi,1  x   η yi,1 
 J         
                 η xi,1  x   η yi,1 
 J     

2Δ η         J  i,3                 i,3  2Δ η
           
        J  i,2             i,2 


2 ρ V                                   1 ρ V 
           (η xi,1 ui,2  η yi,1 v i,2 )           (η xi,1 ui,3  η yi,1 v i,3 )
Δ η  J  i,2                             2Δ η  J  i,3

1  ρU 
            (η x ui,i 1,1  η yi,1 v i 1,1 )
2Δ ξ  J  i 1,1 i,1

1  ρU 
                   (η x ui  1,1  η yi,1 v i  1,1 )
2Δ ξ  J  i  1,1 i,1

When Equation (5.24) is applied to all i at j =1 (Wall of nozzle), the following
tridiagonal system of equations is obtained:

b 2        c2                                    P2,1  d 2  a 2 P1,1           
a                                                                               
 3         b3        c3                          P3,1  d 3                      
                                                                                
                                                                                
                                                                               
                                                                                
                                                                                
          a IMM2      b IMM2        c IMM2       PIMM2,1  d IMM2                
                                                           
                     a IMM1         b IMM1 


 PIMM1,1  d IMM1  c IMM1 PIM,1 
         

PIM,1        PIMM1,1
Since P1,1  P2,1 and                                          . Therefore,
J IM,1       J IMM1,1

114
Chapter Five                                                  Two Dimensional Nozzle Flow

b 2       c2                              P2,1  d 2 
                                                          
a 3      b3       c3                      P3,1  d 3 
                                                          
                                                          
                                                                  ..........        .......... (5.25)
..........        ......
                                                          
                                                          
        a IMM2     b IMM2      c IMM2     PIMM2,1  d IMM2 
                                                          

                   a IMM1      b IMM1 
    PIMM1,1  d IMM1 
         

Where:

b2  a 2  b2

J IM,1
IMM1 = IM – 1, and           bIMM1  bIMM1  c IMM1
J IMM1,1

Velocity components and density are calculated by using the energy
equation with the geometry parameters as follows:
Total enthalpy at the surface is assumed constant. This statement is
expressed mathematically as:

1 2
k et      (u  v 2 )  (ht )wall  Constant           ..........
..........
..........
..........      ........
..........    (5.26)
2

For a perfect gas;

     Pi,1        1 C i2  Di2
k
 (k  1)ρ

 2 ρ2          (ht )wall          ..........        ..........
..........        ..........        .........
..........      (5.27)
          i,1           i,1

This equation is rearranged as:

2 (k  1)(ht )wall ρ i,1  (2k)Pi,1 ρ i,1  (1  k)(Ci2  Di2 )  0.0
2
..........
..........                ..
..........(5.28)

Then the density at the wall is solved as follows:

115
Chapter Five                                                          Two Dimensional Nozzle Flow

2k Pi,1  4k 2 Pi,1  8 (k  1)2 (C i2  Di2 )(h t )wall
2

ρ i,1                                                                         .......... .......... .......... ........ (5.29)
4(k  1)(h t )wall

The positive sign is used exclusively to prevent negative density

values. The velocity at the wall of nozzle may be initialized in different
ways. One way extrapolate U from the interior points, or, the same value

for ρ V is imposed at the wall of nozzle with the vector rotated such that

the velocity is tangent at the surface. Thus,


Ai  ρ V                 (ρ u)2  (ρ v)2           ..........
..........
..........
..........        ..........
..........                 ...
.......... (5.30)
i,2

From the geometry parameters, it is useful to find the component of
velocity u and v as follow:

ηy                  η x
sinθ               , cosθ 
bb               bb

bb  η x  η y
2     2

C i  ρ u i,1  Ai cosθ 

Di  ρ v i,1  Ai sinθ 

All of these quantities are illustrates in Figure (5.2).

η
Ai                                                                116
j=2
Chapter Five                                   Two Dimensional Nozzle Flow

5.3.4     Artificial viscosity.
Numerical dissipation is the key to creating stable and accurate
numerical algorithm. Virtually all computational fluid dynamic techniques
contain artificial viscosity to some degree, either implicitly or explicitly

[26].
The Euler equations omit viscosity; however, the discretization
generally reintroduces viscosity or, more precisely, second difference terms
that has viscous like effects. Second differences that arise naturally as part
of the first derivative approximation are called implicit artificial viscosity.
Second differences purposely added to first derivative approximations

are called explicit artificial viscosity.

117
Chapter Five                                          Two Dimensional Nozzle Flow

Artificial    viscosity       forms   sometimes   suggest   alteration   and
improvements. The implicitly artificial viscosity is too small making
unstable, and adding explicit artificial viscosity with a positive coefficient
has a smoothing and stabilizing effect. In other cases, a numerical method
may have too much artificial viscosity with a positive coefficient has a
smoothing and stabilizing effect. In other cases, a numerical method may

have too much artificial viscosity, causing smearing or even instability. In
this case, adding explicit artificial viscosity with a negative coefficient
partial cancels the implicitly artificial viscosity, resulting in a sharper, and
even more stable solution. Second order artificial viscosity with a positive
coefficient is sometimes called artificial dissipation; second order artificial
viscosity with negative coefficient is sometimes also called artificial anti -
dissipation.
In the current study a four order smoothing term amounts to adding to
the predictor and corrector steps the difference form of the following term:

Predictor step.

U   U   SU 
1
t 1
1
t 1       t 1
1    i, j

Where SU1 is a fourth order artificial viscosity term, defined by:

118
Chapter Five                                                                                           Two Dimensional Nozzle Flow

Pi t 1, j  2Pi,t j  P i 1, tj
(SU   t 1
)i, j  C x                                                           
 (U 1 )it 1, j  2(U 1 )i, j  (U 1 )it1, j
t

 2P  P
1                        t                   t               t
P i  1, j             i, j           i  1, j

Pi,t j  1  2Pi,t j  P i, j - 1
t

C y
Pi,t j  1  2Pi,t j  Pi,t j - 1

 (U 1 )i, j  1  2(U 1 )i, j  (U 1 )i, j 1
t                 t            t
                   .(5.31)
..........

Corrector step.

U   U   SU 
1
t 1
1
t 1                   t 1
1    i, j

Where:


Pi t1, j  2Pi,t j 1  P i t1,1
1

(SU   t 1
)i, j  C x
j

 (U 1 )it 1, j  2(U 1 )i, 1  (U 1 )it1, j
1               t              1

 2P  P
1                          t                     t            t                                       j
P  i  1, j               i, j        i  1, j

Pi,t j11  2Pi,t j 1  P i,tj- 1
C y
P   t 1
 2P         t 1
1

P      t 1

 (U 1 )i, 1  2(U 1 )i, 1  (U 1 )i,1
t 1
j
t
j
t 1
j               .........
(5.32)
i, j  1               i, j           i, j - 1

The fourth order nature of Equations (5.88) and (5.98) can be seen in
the numerators, which are products of two second – order central difference
expressions for second derivatives, many attempts in choosing different

values of coefficient, Cx , and Cy . Typical values of these coefficient range
from (0.01 to 0.3) [22].
A computer flow chart for performing calculation for this method
(Explicit MacCormack’s Technique) is presented in Figure (5.3).

119
Chapter Five                                      Two Dimensional Nozzle Flow

Start

Enter the initial flow data and constants for the physical domain

Specify the geometry of physical domain (nozzle geometry)

Transform the physical domain to computational domain
& Compute the transformation parameters.

Initialize flow field variables p, u, v, etc.

Do
Iter. = 1,Max.Iter.

Compute DT for stability requirement (CFL stability).

Compute flux vectors in terms of U1, U2, U3,
E1, E2, E3, F1, F2, and F3 at all grid points

Predicted step

B
A
Figure (5.3)
Flow chart of MacCormack’s scheme (Explicit solution).

120
Chapter Five                                Two Dimensional Nozzle Flow

B                                       A
Solve finite difference equations to obtain flux term (U)
at internal grid points.

Decoding

Boundary conditions (wall, inflow, downstream)

Compute flux vectors in terms of U1, U2, U3,
E1, E2, E3, F1, F2, and F3 at all grid points

Corrector step

Solve finite difference equations to obtain flux term (U)
at internal grid points.

Decoding

Boundary conditions (wall, inflow, downstream)

End

Continue of :Figure (5.3).

121
Chapter Five                                  Two Dimensional Nozzle Flow

5.4 Numerical Results And Discussion.

The direct problem is usually the one encountered in practice, and the
time dependent approach described here is the only technique available at
present that allows the exact solution of the direct problem. The successful
development of the time dependent technique now provides a uniformly
valid calculation of the complete subsonic- supersonic flow field in a two

dimensional nozzle.
MacCormack’s technique is recommended for the finite difference
calculation of time derivative. After a number of time steps advance from
set of initial values of the flow field at each grid point and prescribed
steady boundary conditions, the flow properties at each grid point will
Figures (5.3) through (5.12) shows the steady state results for two
dimensional supersonic flow nozzle. These are contour and vector plots of
the velocity, density, temperature, pressure, and Mach number. These
results are for two geometry of nozzle; nozzle 1 with geometry (5.95:5.95)
and nozzle 2 with geometry (2.5:1.5). Number of grid was equal to (72) in
x and (21) in y direction for nozzle 1, while it’s equal to (201) in x and (21)
in y direction for nozzle 2. Results are calculated with values of the
dissipation factors Cx and Cy are equal to 0.5.
The transverse velocity (the y component of velocity, v ) do not appear

in one dimensional flow where the properties and velocity was uniform
across the section (Quasi- one dimensional flow) with absence of this
component. But here the effect of not absence of this velocity component is

recognizing on the flow properties. Because of the presence of change in

122
Chapter Five                                  Two Dimensional Nozzle Flow

area in the flow field, the flow field becomes two dimensional with
transverse velocity, along with the various vectors, as can be seen in the
vector plot.
5.4.1 Discussion of results for nozzle 1.

Figure (5.3) display of a vector quantity along the nozzle 1, where the
vector plot is a display of a vector quantity (usually velocity) at discrete

grid points, shown both magnitude and direction, where the base of each
vector is located at the respective grid point as shown in this figure.
Figure (5.4) shows the contour plot of density, from which can be see
that the decreasing of the density value from this value at inlet boundary to
the certain value at the throat of nozzle and to certain continuos decreasing
through the divergent part of nozzle 1. This decreasing in density value is
due to increasing in velocity from subsonic value to supersonic value due
to change in area with conservative of mass flow rate (steady state
condition).
A contour line is a line along which some property is constant.
Generally, contours are plotted such that the difference between the
qualitative value of the dependent variable from one contour line to an
adjacent contour line is held constant.
Figure (5.5) shows the contour lines of temperature. This plot shows the
decreasing of temperature value through the nozzle from this value at inlet
boundary to the certain values at the convergent and divergent parts of
nozzle 1. Increasing in value of velocity causing this decreasing in
temperature value because the alternating between kinetic energy and

123
Chapter Five                                   Two Dimensional Nozzle Flow

internal energy which causing to decrease temperature with increase of
velocity, this due to energy conservation.
Figure (5.6) shows contour plot of pressure along the nozzle 1. The
same behavior of density and temperature can be seeing here; the flow
expanded from inlet pressure point to back pressure value at the exit. This
similarity in behavior is due to state equation and momentum conservation.

Figure (5.7) shows increasing in Mach number value to unity at the
throat of nozzle and to certain values greater than one in divergent part
until the Mach number is arrive to exit Mach number. The unity value of
Mach number do not be uniform along the throat of nozzle, but take the
curve form with many grid points as shown in Figures (5.7). In this figure it
is anticipated that the cross-section of the sonic surface termed the sonic
line, is a parabola; it is the locus of all of the points in the flow field where
the flow Mach number equal to 1. On the wall, at the throat, both the slop
of the wall and y component of velocity are zero, so that the flow velocity
is parallel to the wall. At every other point along the wall, there is some
small difference between the wall angle and the flow angle as shown in
Figure (5.3).
It is clear from examining these contour plots that the global nature of
the two dimensional flow is seen in one single view; to obtain the same
global feeling for the results from x-y plots, say ascertain the flow field
properties. Contour plots are clearly a superior graphical representation
from this point of view.
From contour plots, the one can be observe that the sharp change in

diffusion angle (wall angle) causing to appearance a large change in

124
Chapter Five                                   Two Dimensional Nozzle Flow

properties value along the certain section, where the constant property
contour lines take the sharp and closer curve form. Because this sharp
change in wall angle causing the rapidly changing in the flow field variable
where the adjacent contour lines are closely spaced together in regions
when the dependent variable is rapidly changing in space Anderson [22].

5.4.2 Discussion of results for nozzle 2.

Figure (5.8) shows the vector quantity along the nozzle 2, this figure
also show that both the slop of the wall and y component of velocity are
zero on the wall at the throat of nozzle, so that the flow velocity is parallel
to the wall. Also, at every other point along the wall, there is some small
difference between the wall angle and the flow angle as shown in Figure
(5.8).
Figure (5.9) shows the contour lines of density from which can be see
that the decreasing of the density value from this value at inlet boundary to
the certain value at the throat of nozzle and to certain continuos decreasing
through the divergent part of nozzle 2. This decreasing in density value as
indicated previously is due to increasing in velocity from subsonic value to
supersonic due to change in area value with conservative of mass flow rate.
Figure (5.10) shows the contour lines of temperature. This plot shows
the decreasing of temperature value through the nozzle 2 from this value at
inlet boundary to the certain values at the convergent and divergent parts of
nozzle. Also the alternating between kinetic energy and internal energy
which is causing to decrease temperature with increase of velocity, this due

to energy conservation.

125
Chapter Five                                   Two Dimensional Nozzle Flow

Figure (5.11) shows contour plot of pressure along the nozzle 2. Also as
in nozzle 1 the same behavior of density and temperature can be seeing
here; the flow expanded from inlet pressure point to back pressure value at
the exit. This similarity in behavior is due to state equation and momentum
conservation.
Figure (5.12) shows increasing in Mach number value to unity at the

throat of nozzle and to certain values greater than one in divergent part
until the Mach number is arrive to exit Mach number. Here, also the unity
value of Mach number do not be uniform along the throat of nozzle, but
take the curve form with many grid points as shown in Figures (5.12). This
result can be shown in Figure (5.13) that obtained from Serra [44] which
also show that the Mach number does not be uniform across the throat, but
take the curve form. In Figure (5.12) it is anticipated that the cross-section
of the sonic surface termed the sonic line, is a parabola; it is the locus of all
of the points in the flow field where the flow Mach number equal to 1. On
the wall, at the throat, both the slop of the wall and y component of velocity
are zero, so that the flow velocity is parallel to the wall. At every other
point along the wall, there is some small difference between the wall angle
and the flow angle as shown in Figure (5.8).
It is clear from examining these contour plots in Figures (5.9) through
(5.12) that the global behavior of properties for two-nozzle geometry’s is
differs from nozzle to another. Here the contour lines of properties was
uniform distribution as straight lines as approximately where the change in
diffusion angle was decreasing slightly and step by step as shown in

figures, where the sharp change in properties do not occur here. But in

126
Chapter Five                                 Two Dimensional Nozzle Flow

contrast, the adjacent contour lines are widely spaced and the dependent
variable is slowly changing in space. This behavior may be due to
increasing in y component of velocity value when the change in diffusion
angle was sharp, which leads to sharp change in resultant velocity. And
what reinforces this conclusion, the contour lines at convergent part of
nozzle was less curvature than these curves at divergent part, where the
value of v in convergent part less than it’s value in divergent part, because

the velocity is subsonic at convergent part, while it’s supersonic in
divergent part. Therefore, the conformability of sharp change in diffusion
angle with increase of v value may be validating the present work’s

conclusion that demonstrated above.
5.4.3 Comparison the results for one and two dimensional flow

The results in Figures (5.3) through (5.12) differ from these results
that obtained by one dimensional flow work, and this difference in values
was demonstrated in Table (5.1) which include the flow properties in one
and two dimensional flow at the throat and exit of nozzle as well as the
values of flow properties for a select interior grid point such as x=7.5, and
x=2. The difference between of them is due to appearance of the y
component of velocity v which do not appear in one dimensional flow

where the properties and velocity was uniform across each section with
absence of this component. The effect of this velocity component is
recognizing on the flow properties. Here, the axial velocity component is
less than of axial velocity for one-dimensional flow because of the
appearance of y component of velocity v. This leads the properties at to be

127
Chapter Five                               Two Dimensional Nozzle Flow

non-uniform along the certain section as shown in contour plots, and differ
from these for one dimensional flow.

128

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