Computing 3D Nozzle Flows

ISSN: 2395-0560 International Research Journal of Innovative Engineering

www.irjie.com Volume1, Issue 6 of June 2015

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Computing 3D Nozzle Flows

Johannes V. Soulis1, Modestos A. Loukas2

1Fluid Mechanics/Hydraulics Division, Department of Civil Engineering, Demokrition University of Thrace, Xanthi 67100, HELLAS, 2Fluid Mechanics/Hydraulics Division, Department of Civil Engineering, Demokrition University of Thrace, Xanthi 67100, HELLAS

Abstract A robust time-marching finite-volume numerical procedure is presented for three-dimensional, steady, inviscid and incompressible nozzle flows written in strong conservation form. The code, named D3flow, is applied to the conservative form of the Euler equations written in general fully-curvilinear discretization approach. Euler equations through internal flow passages are a good first approximation for analysis. A simple but computational efficient grid is constructed. An explicit marching in time numerical procedure is applied for time integration requiring minimal and straightforward algorithm coding. Numerical solution results and comparisons with a numerical technique (Fluent code) are presented for a 3D nozzle exhibiting complex topological structure of the water flow. Predicted results using either method yield satisfactory comparison. The proposed numerical method is an accurate and reliable technique for solving inviscid, incompressible water flow equations in 3D internal configuration geometries widely used in a turbo machinery environment.

Keywords Euler Equations, Time-marching, Finite-Volume Scheme, Water Flow, 3D Nozzle

1. Introduction Internal configuration geometry development of either turbines or pumps has been largely conditioned by the

improvements achieved in component efficiencies. Design and performance estimation of turbomachines has been based and will continue to be so, almost completely, on the understanding of fluid flow behavior within their passages. However, flow passages through turbomachines are geometrically very complex. Furthermore, the interaction between rotors and stationary parts constitutes a major challenge. Computational fluid dynamics methodology applied to internal flows has been achieved a considerable progress in the past several years. Although efficient algorithms are now available to integrate the Navier-Stokes equations, this appears to be still a formidable task for purposes of practical applications. In the complex machine environment the prediction accuracy of such flow calculations is limited by the limitations of turbulence modeling. Mixing length eddy viscosity models are by far the most commonly used method [1]. The solution of inviscid flows, now Euler equations, through internal flow passages is a good first approximation for analysis. It is of practical interest for the design of turbo machinery components. Inviscid flow equations are numerically treated in two distinct categories, namely Euler solvers and potential flow (irrotational) solvers. Potential methods do not appear to have been widely used for design purposes. Nowadays, 3D Euler solvers are well developed and are available for routine calculations. Several of these dealing with internal and external flows are described in [2]. Euler solvers for 3D turbo machinery flows have been reported as early as 1974 by Denton [3], who developed an explicit time-marching method. His widely accepted method employs an opposed difference scheme in order to solve the Euler equations. The scheme uses upwind differencing for fluxes of mass and momenta, but downwind differences for pressures in the streamwise direction. In addition, correction factors for each of the physical quantities are applied in the streamwise direction. The method is of the finite-volume type. Numerical solution technique was described in [4]. The hop-scotch scheme was applied to the conservative form of the Euler equations written in general curvilinear co-ordinates using an O-type grid system. Researchers in [5] presented a 3D Euler analysis on a C-type grid using the well-known Beam-Warming implicit algorithm. Results for a cascade and rotor flows were presented. An inviscid flow solution for axial turbine stage was presented [6]. A 3D potential flow solver for turbo machinery blade rows was described in [7].

The objective of this paper is to outline an accurate and efficient numerical procedure for simulating the time-averaged, 3D, inviscid water flow field within a typical nozzle exhibiting complex geometry. The main scope was to develop a numerical code capable for computing flows through all types of turbomachines (axial, mixed, radial) no matter how complex their geometry may be and to compare the results with the well-known numerical technique Fluent.



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The proposed scheme has several advantages: The grid used is the simplest possible formation for numerical calculations. The conservative form of the equations is written in general curvilinear co-ordinates, thus enabling complex geometry to be efficiently analyzed. Calculated water mass flows into and out of the nozzle are matched. Boundary conditions are easily and accurately satisfied in a straight forward manner. The time integration numerical procedure is a straightforward method (time marching) requiring minimal algorithm coding.

For the time being applications are restricted to water flow through 3D complex geometry nozzles. The code development was performed at the Computation Laboratory of the Fluid Mechanics/Hydraulics Division, Civil Engineering Department, Demokrition University of Thrace, Hellas.

2. Methodology 2.1. Geometry

The geometry of the tested 3D nozzle is shown in Fig. 1a (side view) and Fig. 1b (top view). a)

b)

Figure 1. Geometry of the 3D nozzle (distorted) a) side view, b) top view (flow from left)

2.2. Mesh generation A complex grid system was developed and implemented on the main numerical method to allow flows inside complex

geometry to be calculated. The tested complex geometry nozzle is shown in Fig. 2a. Hub and casing radial co-ordinates along the machine axis are provided as an input to the computer code.

(r) tangential distance

r radial distance

(z) Axial distance




b) a)

Input cross-sections are interpolated to calculate the axial, suction surface and blade thickness. Meridional grid points are uniformly spaced in the r-direction. Axial points need not be equally spaced. Fig. 2b shows the 3D grid used for nozzle analysis using Fluent [8]. Use of pitch wise lines greatly simplifies the application of the periodic boundary properties between the bounding quasi-streamlines of the passage. However, the numerical scheme can be used with any grid formation, which needs not be uniformly spaced in any co-ordinate direction. The grid is not restricted to the one described. Any mesh generation technique can be adopted. Edges require special numerical treatment. The problem can be overcome by fitting more grid points where appropriate. The D3flow program utilizes 2800 nodes while the Fluent 53361 nodes for the under consideration nozzle flow problem.

Figure 2. a) 3D nozzle geometry, b) mesh used for analysis (Fluent)

2.3. Governing Flow Equations The basic equations governing the flow inside of any internal configuration geometry are derived from the principles of

conservation of mass, momentum and energy. To extend the code for turbo machinery flow analysis it is convenient to write the 3D Euler equations in a cylindrical polar co-ordinate system. Since the algorithm was developed for compressible fluid analysis, it is convenient to write down the equations in compressible flow taking into account the rotational speed . These equations are expressed in conservation form [9] as,

,)()]([)()(rwrr

zur

tr

(1)

,)(])([)]([)(2

ruwrur

zpur

tur

(2)

,)(])([)()( wr

wrpurzur

tr

(3)

),()]([])([)()( 22

pr

pwrwrzuwr

twr

(4)

,])([)])[(])([)(

rwperrepe

zuper

ter

(5)

z, and r are the axial, tangential and radial directions respectively, t is the time, is the density, u, and w are the absolute velocity components along the z, and r directions respectively, p is the pressure of the fluid, is the rotational speed of the impeller and e is the total internal energy given by the equation below,

),(21

)1(222 wupe

(6)




is the ratio of the specific heats. The unknowns of the problem are the six physical quantities u, , w, , p and e. The energy Eq. (5) coupled with Eq. (6) are easily accomplished for water flow.

For current application the gravity acceleration g has negligible effects upon the solution. However, the gravity has been included in all calculations (not shown in Eqs (2)-(4)). Furthermore, the flow passages (nozzles) are not rotating machine components. Thus, the rotational speed is set equal to zero.

2.4. Transformation Equations The discrete approximation to the governing flow equation in compressible flow and rotational framework form has

been developed by dividing the physical domain into cuboid cells which can be defined arbitrarily to produce surface-fitted grids, the structure of which follows the machine internal configuration. Once this has been achieved, a transformation is introduced through which cuboids of the physical domain are mapped into computational domain cubes, Fig. 3, [10].

Figure 3. Distorted cubes (left) of the physical domain are mapped into cubes (right) of the computational domain

The transformation from global z, , r to local , , co-ordinates can be expressed as,

,8

1 i ii zNz ,8

1 i iiN ,8

1 i ii rNr (7) Ni are the first-order, linear shape functions associated with the cuboid nodes. The use of first-order shape functions has been determined by the necessity to restrict the complexity of the numerical code, which is inherent to almost all 3D computational methods. Thus, in order to numerically solve the system of governing flow Eqs (1)(6) on a body-fitted grid system, the equations are transformed to an arbitrary curvilinear system [9],

,)()()()(1111

WrJVrJUrJt

rJ (8)

,)]([)]([)]([)(1111

puWrJpuVrJpuUrJt

urJ zzz (9)

1 11

11

[ rJ ( U p / r )] [ rJ ( V p / r )]( rJ )t

[ rJ ( W p / r )] J W ,

(10)

1 11

r r

11 2r

[ rJ ( wU p )] [ rJ ( wV p )]( rJ w)t

[ rJ ( wW p )] J ( p ),

(11)




1 11

1

{ rJ [( e p )U p( r ) / r )]} { rJ [( e p )V n p( r ) / r )]}( rJ e )t

{ rJ [( e p )W p( r ) / r )]} ,

(12)

U, V and W are the contravariant velocity components in the , and directions, respectively. The inverse Jacobian J-1 of the transformation from physical to local co-ordinate system is defined as,

.1

rrr

zzzJ (13)

The metrics z, z and z, of Eq. (9) are,

,/)( 1 Jrrz ,/)(1 Jrrz

1z ( r r ) / J

(14) similarly, for the metrics , and , as well as for r, r and r. The contravariant velocities are related to the physical velocities by the equations,

,))(( wrr

uU rz

,))(( wrr

uV rz

.))(( wrr

uW rz

(15) 2.5. Flow Discretization A numerical algorithm, D3flow, has been developed to solve the governing flow Eqs (8)-(12), [9]. For a control volume V, shown in Fig. 4 and for a given time step t these equations for the n+1 (current) time iteration step may be written as,

n1 n 1 1 1 1( rJ ) ( rJ U ) ( rJ V ) ( rJ W ) t / continuity (16)

n1 n 1 1 1 1x x x( rJ u ) rJ ( uU p ) rJ ( uV p ) rJ ( uW z p ) t / -momentum (17)

The -momentum, -momentum and energy equations are formed accordingly. For each of the continuity, -momentum (axial, X), -momentum (tangential, T), -momentum (radial, R) and energy equations the XFLUX, TFLUX and RFLUX quantities are formed (the example given refers to -momentum fluxes) as,

Figure 4. Notation and flux balancing across a finite-volume (===unit)




1 1 1i , j ,k x x xi , j ,k i , j 1,k i 1, j ,k 1

1x i , j ,k 1

( XFLUX ) { rJ ( uU p ) rJ ( uU p ) rJ ( uU p )

rJ ( uU p ) }

-momentum flux (18)

The differences appearing in the RHS of Eqs (16) and (17) as well as for the rest of the transformed Euler equations are defined as follows (the example given refers to -momentum flux),

1x i, j 1,k i , j ,ki, j ,k

rJ ( uU p ) ( XFLUX ) ( XFLUX )

1x i 1, j ,k i 1, j ,ki , j ,k

rJ ( uV p) (FLUX ) (FLUX ) 0.5

1x i, j ,k 1 i ,j ,k 1i , j ,k

rJ ( uW p) (RFLUX ) (RFLUX ) 0.5 (19)

Backward finite differences are used for the partial derivatives of z, , r with each of , , respectively. Now all fluxes and partial derivatives may be used in Eqs (16) and (17) and the rest equations to obtain the , u, v, w and e via 1( rJ ) n+1,

1 n 1( rJ u ) , 1( rJ v ) n+1 1( rJ w ) n+1, 1 n 1( rJ e ) . The procedure is,

1 n 1

n 1 n i , j ,k1 11 n 1i, j ,k i , j ,k

i , j ,k

( rJ )rJ rJ

1.0 0.0001 ( rJ )

continuity (20)

1 n 1

n 1 n i, j ,k1 11 n 1i, j ,k i, j ,k

i, j ,k

( rJ u )rJ u rJ u

1.0 0.000025 ( rJ u )

momentum (21)

Similar action is taken for the momentum, momentum and energy equations. Once pressures are calculated via the energy equations, new values for pressure are calculated as follows: a) updated pressures are taken from the downwind face of the cell and at the same time they are corrected according to,

i , j ,k i ,j 1,k i , j 1,k i , j 1,k

n 1 n 1 n 1 n 11p p C ( p p )0.5

(22)

b) pressures are applied to each of the momenta equations and c) pressures are relaxed before they are incorporated in the Eq. (22). A typical value for C1 is 0.1. The XFLUX quantities are expressed in upwinding differencing while the TFLUX and RFLUX quantities remain unaltered. Thus, the proper expression for Eq. (18) becomes,

1 1 1i , j ,k x x xi , j 1,k i , j ,k i 1, j 1,k 1

1x i, j 1,k 1

( XFLUX ) { rJ ( uU p ) rJ ( uU p ) rJ ( uU p )

rJ ( uU p ) }

(23)

2.6. Flow Boundary Conditions A complete specification of the boundary conditions is needed to close the problem. Upstream flow conditions: The stagnation pressure p01 (Pa) at the inlet flow (as well as at any place within the flow) is assumed to be constant. No velocity angle is imposed. Downstream flow conditions: The static pressure p2 (Pa) is set to a constant value on the hub surface and the radial pressure distribution is determined by the simple radial equilibrium equation,

2p wr r

(24)




Body surface conditions: In inviscid flow analysis the flow is tangent to a wall surface, so the flow normal to the suction or pressure and/or to the endwall surfaces is zero. In nozzle flow where highly loaded surfaces and curved hub or casing surfaces are frequently encountered, the accurate description of the wall boundaries is paramount importance in obtaining satisfactory results. To achieve the most favourable comparisons between applied methods, the outlet D3flow boundary conditions are accordingly adjusted. The flow boundary conditions as well as the flow physical quantities for the 3D nozzle problem using D3flow and Fluent technique are shown in Table 1.

Table 1. Boundary Conditions and Flow Physical Quantities Used for D3flow and Fluent Codes

Fluent D3flow static pressure at inlet p1

(initial value) 140000.0 Pa 140000.0 Pa

static pressure at outlet p2 101000.0 Pa 101000.0 Pa

total pressure at inlet p01 144000.0 Pa 144000.0 Pa velocity wall

boundaries u, v, w solid (perpendicular to

wall velocities are set zero) solid (perpendicular to

wall velocities are set zero)

fluid density (water) 1000.0 kg/m3 1000.0 kg/m3

rotational velocity 0.0 rad/s 0.0 rad/s

gravity acceleration g 9.81 m/s2 9.81 m/s2

2.7. The Iterative Scheme The total number of iterations required to achieve convergence strongly depends on the actual time-step. The

convergence criterion based on average error for each flow variable i.e. velocities, pressure and continuity for Fluent solution was set to 10-6. Fluent used 53361 nodes to achieve the above mentioned convergence criterion.

As with all marching in time methods, as the D3flow code is, the theoretical maximum stable time step t is specified according to the Courant-Friedrichs-Lewy (CFL) criterion. Grid reduction tests for the D3flow code have shown that the grid size, i.e. the ratio of x to y or z, does not affect the accuracy of the solution. In extreme ratios the solution breaks down. The D3flow method is an explicit marching in time numerical technique requiring substantial computational time to achieve solution. However, the amount of nodes needed to satisfactorily converge i.e. error dropping below 10-6, is 2800 nodes only. Convergence is achieved when all physical quantities (velocities plus mass balance) reaches the above mentioned convergence criterion. 3. Computational Results and Discussion

The current research work presents the results of solving the 3D nozzle problem using Fluent and D3flow. The computational static pressure (N/m2 or Pa) distribution comparison between the methods D3flow and Fluent are shown in Figs 5, 6, 7 and 8 along bottom-right side, upper-right side, bottom-left side and upper-left side, respectively.

Figure 5. Static pressure (Pa) comparison between D3flow

(circles)Fluent (squares) along bottomright side

Figure 7. Static pressure (Pa) comparison between D3flow (circles)Fluent (squares) along bottom-left side




Figure 6. Static pressure (Pa) comparison between D3flow (circles)Fluent (squares) along upperright side

Figure 8. Static pressure (Pa) comparison between D3flow (circles)Fluent (squares) results along upperleft side

The computational total velocity (m/s) distribution comparison between the D3flow and Fluent are shown in Figs 9, 10, 11 and 12 along bottom right-side, upper-right side, bottom-left side and upper-left side, respectively.

0.00 m s-1

2.00 m s-14.00 m s-16.00 m s-1

8.00 m s-110.00 m s-1

12.00 m s-114.00 m s-1

0.00 m 0.10 m 0.20 m 0.30 m

Figure 9. Velocity magnitude (m/s) comparison between D3flow (circles)Fluent (squares) along bottomright side

0.00 m s-12.00 m s-14.00 m s-16.00 m s-18.00 m s-1

10.00 m s-112.00 m s-114.00 m s-1

0.00 m 0.10 m 0.20 m 0.30 m

Figure 11. Velocity magnitude (m/s) comparison between D3flow (circles)Fluent (squares) along bottomleft side

4.00 m s-16.00 m s-18.00 m s-1

10.00 m s-112.00 m s-114.00 m s-116.00 m s-118.00 m s-1

Figure 10. Velocity magnitude (m/s) comparison between D3flow (circles)Fluent (squares) along upperright side

0.00 m s-1

2.00 m s-1

4.00 m s-1

6.00 m s-1

8.00 m s-1

10.00 m s-1

12.00 m s-1

0.00 m 0.10 m 0.20 m 0.30 m

Figure 12. Velocity magnitude (m/s) comparison between D3flow (circles)Fluent (squares) along upperleft side

z, Axial distance




From the nozzle geometry shown in Fig. 1, it is clear that the r axis attains the height of 50.0 mm at inlet (z=0.0 mm) and keeps it up to z=19.0 mm. Thereafter, the geometry converges up to z=75.0 mm, where a throat is formed. Subsequently, the geometry divergence ends at z=150.0 mm. From this point down to nozzle exit the height attains 60.0 mm.

Reduction in the static pressure (Pa) occurs from inlet up to the distance of z=120.0 mm (throat) at the bottom-right side of the nozzle. Thereafter, the static pressure increases, Fig. 5. This trend is captured by both numerical methods. At the bottom-left side, Fig. 7, reduction in the static pressure occurs from inlet up to the distance of z=140.0 mm. Thereafter, the pressure increases. The two numerical methods agreement is good. Contours of static pressure at the bottom side using the Fluent code are shown in Fig. 14. At the nozzle upper-right side the static pressure attains its lowest value at axial distance z=75.0 mm, Fig. 6. The pressure drop is high. However, both numerical methods satisfactorily capture this drop. Thereafter, the static pressure is recovered and this is well captured by the applied numerical techniques. At the upper-left side, a drop of static pressure also occurs reaching its lowest value at z=75.0 mm, Fig. 8. However, the low static value for this side is considerable milder to the one appearing in upper-right side, Fig. 7. There is pressure recovery downstream to z=75.0 mm. The pressure keeps the value attained at the throat of the nozzle for some distance downstream to the throat. Contours of static pressure at the upper side of nozzle using Fluent are shown in Fig. 13. The low static pressure region is clearly depicted and it is shown in blue colour.

The velocity (m/s) distribution is almost inversely proportional to the static pressure changes, Figs. 9, 10, 11 and 12. The velocity comparison between the two numerical techniques is satisfactory. Some comparison differences occurring are attributed to the geometry description irregularities. The nozzle, as it is expected, accelerates the water flow velocity from nearly 5.0 m/s at nozzle inlet to 9.0 m/s at nozzle outlet.

Figure 13. Fluent computed contours of static pressure (Pa) upper-side (casing) view

Figure 14. Fluent computed contours of static pressure (Pa) bottom-side (hub) view




4. Conclusion An accurate and efficient numerical method, D3flow code, for three-dimensional, inviscid, incompressible analysis of

internal configuration flows using general curvilinear coordinates has been developed and its ability to simulate complex geometries was demonstrated. The technique is a straight forward explicit marching in time numerical scheme requiring minimal algorithm coding. Numerical solution results and comparisons with available numerical solution of Fluent code have been presented for a nozzle exhibiting 3D flow structure. The computational results comparison between the applied codes is satisfactory. The proposed code is based on a simple body-conforming grid system. It utilizes a simple time integration technique. Main features of the flow are reasonably well predicted by both codes, even using comparatively coarse grids for the D3flow.

REFERENCES [1] Denton, J. D. (1990), The calculation of three-dimensional viscous flow through multistage turbomachines, ASME Paper, 90-

GT-19. [2] Hirsch, C. (1990), Numerical computation of internal and external flows, Vol. 2, Wiley, Chichester, Chap. 16, pp. 132-583. [3] Denton, J. D. (1974), A time-marching method for two-dimensional and three-dimensional blade to blade flows, ARC R&D

3775. [4] Shieh, C. F., Delaney, R. A. (1986), An accurate and efficient Euler solver for three-dimensional turbo machinery flows, ASME

Paper, 86-GT-200. [5] Weber, K. F. , Thoe D. W., Delaney, R. A. (1990), Analysis of three-dimensional turbo machinery flows on C-type grids using

an implicit Euler solver, Journal Turbo machinery, 112, pp. 362-369. [6] Arts, T. (1985), Calculation of the three-dimensional steady inviscid flow in a transonic axial turbine stage, ASME Paper, 84-

GT-76. [7] Soulis, J. V. (1983), Finite-volume method for three-dimensional transonic potential flow through turbo machinery blade rows,

Int. J. Heat Fluid Flow, 4. [8] ANSYS Inc., (2012) Ansys Fluent Theory Guide, Release 14.0. [9] Soulis, J. V. (1995), An Euler solver for three-dimensional turbo machinery flows, International Journal for Numerical Methods

in Fluids, vol.20, pp. 1-30. [10] Soulis, J. V. , Bellos, K. V. (1988), Conservation form of fluid dynamics equations in curvilinear coordinate systems, Part I,

Mathematical analysis, Tech. Chron. B, 8, (4), pp. 69-97.

Computing 3D Nozzle Flows

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