implicit turbine cascade cfd solver

7/26/2019 Implicit Turbine Cascade CFD SOlver

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Implicit numerical simulation of transonic flowthrough turbine cascades on unstructured gridsY Meiand A Guha

Aerospace Engineering Department, University of Bristol, Bristol, UK

The manuscript was received on 4 August 2004 and was accepted after revision for publication on 26 August 2004.

DOI: 10.1243/095765005X6926

Abstract: Numerical simulation of the compressible flow through a turbine cascade is studiedin the present paper. The numerical solution is performed on self-adaptive unstructured meshesby an implicit method. Computational codes have been developed for solving Euler as well asNavier Stokes equations with various turbulence modelling. The Euler and Navier Stokescodes have been applied on a standard turbine cascade, and the computed results are compared

with experimental results. A hybrid scheme is used for spatial discretization, where the inviscidfluxes are discretized using a finite volume method while the viscous fluxes are calculated bycentral differences. A MUSCL-type approach is used for achieving higher-order accuracy. Theeffects of the turbulent stress terms in the Reynolds-averaged NavierStokes equations havebeen studied with two different models: an algebraic turbulence model (Baldwin Lomaxmodel) and a two-equation turbulence model (kvmodel). The system of linear equations issolved by a GaussSeidel algorithm at each step of time integration. A new treatment of thenon-reflection boundary condition is applied in the present study to make it consistent withthe finite volume flux calculation and the implicit time discretization.

Keywords: upwind, CFD, computation, time-marching, Euler, Navier Stokes, unstructuredgrid, self-adaptive, turbulence, turbine, implicit

1 INTRODUCTION

Computational fluid dynamics (CFD) simulation hasbecome an essential tool in the design and analysisof modern turbomachinery components during thepast decade. Steady and unsteady state flow predic-

tions are widely studied for problems ranging insize from a single compressor or turbine blade to acomplete multistage turbomachine. CFD plays anincreasingly important role for it offers the followingadvantages.

1. Improved designs:(a) better understanding of flow from CFD leads

to improved designs;(b) higher efficiency, lower losses, and wider

operating characteristics.2. More reliable design methods:

(a) More accurate estimates of performanceduring the design process;

(b) less reliance on empirical database and expertknowledge.

3. Quicker development delivery times:(a) design in virtual reality requires less hardware

and fewer prototypes to be made.4. Less expensive design process:

(a) reduction in the number of expensive rig andprototype tests;

(b) better understanding when something goeswrong.

The continual increases in engine pressure ratio andmaximum temperature require accurate predictionsof the aerodynamic characteristics and of the heatloads imposed on the blades.

The final purpose of the present research is toinvestigate the complex physical phenomenon

when non-equilibrium wet steam flows through a

low-pressure steam turbine. As the first part, the pre-sent paper is specifically concerned with dry gaspredictions for cascade flow.

An implicit time integration is used here for betternumerical stability and convergence. In the past,

Corresponding author: Aerospace Engineering Department,

University of Bristol, University Walk, Bristol BS8 1TR, UK.

35

A09404 IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

Implicit numerical simulation of transonic flowthrough turbine cascades on unstructured gridsY Meiand A Guha

Aerospace Engineering Department, University of Bristol, Bristol, UK

The manuscript was received on 4 August 2004 and was accepted after revision for publication on 26 August 2004.

DOI: 10.1243/095765005X6926

Abstract: Numerical simulation of the compressible flow through a turbine cascade is studiedin the present paper. The numerical solution is performed on self-adaptive unstructured meshesby an implicit method. Computational codes have been developed for solving Euler as well asNavier Stokes equations with various turbulence modelling. The Euler and Navier Stokescodes have been applied on a standard turbine cascade, and the computed results are compared

with experimental results. A hybrid scheme is used for spatial discretization, where the inviscidfluxes are discretized using a finite volume method while the viscous fluxes are calculated bycentral differences. A MUSCL-type approach is used for achieving higher-order accuracy. Theeffects of the turbulent stress terms in the Reynolds-averaged NavierStokes equations havebeen studied with two different models: an algebraic turbulence model (Baldwin Lomaxmodel) and a two-equation turbulence model (kvmodel). The system of linear equations issolved by a GaussSeidel algorithm at each step of time integration. A new treatment of thenon-reflection boundary condition is applied in the present study to make it consistent withthe finite volume flux calculation and the implicit time discretization.

Keywords: upwind, CFD, computation, time-marching, Euler, Navier Stokes, unstructuredgrid, self-adaptive, turbulence, turbine, implicit

1 INTRODUCTION

Computational fluid dynamics (CFD) simulation hasbecome an essential tool in the design and analysisof modern turbomachinery components during thepast decade. Steady and unsteady state flow predic-

tions are widely studied for problems ranging insize from a single compressor or turbine blade to acomplete multistage turbomachine. CFD plays anincreasingly important role for it offers the followingadvantages.

1. Improved designs:(a) better understanding of flow from CFD leads

to improved designs;(b) higher efficiency, lower losses, and wider

operating characteristics.2. More reliable design methods:

(a) More accurate estimates of performanceduring the design process;

(b) less reliance on empirical database and expertknowledge.

3. Quicker development delivery times:(a) design in virtual reality requires less hardware

and fewer prototypes to be made.4. Less expensive design process:

(a) reduction in the number of expensive rig andprototype tests;

(b) better understanding when something goeswrong.

The continual increases in engine pressure ratio andmaximum temperature require accurate predictionsof the aerodynamic characteristics and of the heatloads imposed on the blades.

The final purpose of the present research is toinvestigate the complex physical phenomenon

when non-equilibrium wet steam flows through a

low-pressure steam turbine. As the first part, the pre-sent paper is specifically concerned with dry gaspredictions for cascade flow.

An implicit time integration is used here for betternumerical stability and convergence. In the past,

Corresponding author: Aerospace Engineering Department,

University of Bristol, University Walk, Bristol BS8 1TR, UK.

35



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various upwind schemes [1 3] have been used forcalculation of the flux, while the StegerWarmingscheme [4] has been used for treating the non-reflecting boundary conditions and also for genera-ting the implicit Jacobian (when implicit timeintegration was involved). In the present work, anew method is developed such that the calculation

of the flux, the boundary conditions, and the implicitJacobian all conform to the Roe scheme. Thisprovides internal consistency and improves theconvergence significantly.

A hybrid scheme is used for spatial discretization,where the inviscid fluxes are discretized using afinite volume method while the viscous fluxesare calculated by central differences. A MUSCL-type approach is used for achieving higher-orderaccuracy. The effects of the turbulent stress termsin the Reynolds-averaged NavierStokes equations

have been studied with two different models: theBaldwinLomax algebraic turbulence model [5] andthe two-equation kv turbulence model [6]. Thesystem of linear equations is solved by a GaussSeidel algorithm at each step of time integration.

Unstructured grids are most suitable for complexand irregular geometries. There are two majorunstructured grid generation methods: the Delaunay

Voronoi method (DVM) and the advancing frontmethod (AFM) for triangles in two dimensions. Bothof the methods are used in the present work, the initial

grids are generated for inviscid and viscous compu-tations by DVM and AFM while the self-adaptive -procedure is based on the DVM theory.

2 GOVERNING EQUATIONS

The equations used to model the flow are the com-pressible, Reynolds-averaged continuity, momen-tum, and energy equations written in an integral

form, where the volume of a computational cell isdenoted byVand its surface byS

V

@W

@t dV

S

FdS

S

G dS0 (1)

where

W r, ru, eT

F r(un), ru(un)pn, (ep)(un)T

G 0, tS, tS u_qqST

tS

txx txy

txy tyy

" #n, _qqS

_qqx

_qqy

24

35 n

By assuming that the fluid is a perfect gas, thepressurep is calculated from

p(g1)(e12 kuk2)

The shear stress tensor, tij, is given by

tijm @ui@xj

@uj@xi

23

@uk@xk

dij

The above system of equations needs initial con-ditions prescribing the flow state att 0 and bound-ary conditions. In the time marching method thefinal steady state is achieved as the convergedsolution of unsteady calculations.

3 NUMERICAL METHOD

The governing equations are treated in conservativeform and discretized in time using the Euler implicitmethod leading to a set of non-linear finite differenceequations, which are solved using a Newtonprocedure. In stationary simulations, convergencehistory is accelerated by using local time steps

which are based on local stability criteria.Applying Greens theorem, the NavierStokes

equations can be rewritten in a differential form

@W

@t r F r G (2)

3.1 Inviscid fluxes

The convective (Euler) parts are discretized using athird-order accuracy, TVD-upwind finite volumescheme. When solving transport equations for turbu-lent quantities, a more stable first-order upwindscheme is used. The TVD scheme is based on theMUSCL-type of upwind scheme [7] which consists

of a projection stage and an evolution stage. In theprojection stage, left and right states at each controlvolume interface are determined by extrapolating thenode values of the conservative variables towardsthe control volume interface. In the evolution stagethe inviscid flux is evaluated by solving the Riemannproblem between left and right states using Roesapproximate Riemann solver [2].

3.2 Viscous fluxes

In order to construct the numerical viscous flux

vector at the control volume interfaces, it is neces-sary to evaluate first-order derivatives of the velocitycomponents, the speed of sound, and the turbulentquantities, which is done in a central-differencesmanner, using Greens theorem.

36 Y Mei and A Guha

Proc. IMechE Vol. 219 Part A: J. Power and Energy A09404 IMechE 2005

various upwind schemes [1 3] have been used forcalculation of the flux, while the StegerWarmingscheme [4] has been used for treating the non-reflecting boundary conditions and also for genera-ting the implicit Jacobian (when implicit timeintegration was involved). In the present work, anew method is developed such that the calculation

of the flux, the boundary conditions, and the implicitJacobian all conform to the Roe scheme. Thisprovides internal consistency and improves theconvergence significantly.

A hybrid scheme is used for spatial discretization,where the inviscid fluxes are discretized using afinite volume method while the viscous fluxesare calculated by central differences. A MUSCL-type approach is used for achieving higher-orderaccuracy. The effects of the turbulent stress termsin the Reynolds-averaged NavierStokes equations

have been studied with two different models: theBaldwinLomax algebraic turbulence model [5] andthe two-equation kv turbulence model [6]. Thesystem of linear equations is solved by a GaussSeidel algorithm at each step of time integration.

Unstructured grids are most suitable for complexand irregular geometries. There are two majorunstructured grid generation methods: the Delaunay

Voronoi method (DVM) and the advancing frontmethod (AFM) for triangles in two dimensions. Bothof the methods are used in the present work, the initial

grids are generated for inviscid and viscous compu-tations by DVM and AFM while the self-adaptive -procedure is based on the DVM theory.

2 GOVERNING EQUATIONS

The equations used to model the flow are the com-pressible, Reynolds-averaged continuity, momen-tum, and energy equations written in an integral

form, where the volume of a computational cell isdenoted byVand its surface byS

V

@W

@t dV

S

FdS

S

G dS0 (1)

where

W r, ru, eT

F r(un), ru(un)pn, (ep)(un)T

G 0, tS, tS u_qqST

tS

txx txy

txy tyy

" #n, _qqS

_qqx

_qqy

24

35 n

By assuming that the fluid is a perfect gas, thepressurep is calculated from

p(g1)(e12 kuk2)

The shear stress tensor, tij, is given by

tijm @ui@xj

@uj@xi

23

@uk@xk

dij

The above system of equations needs initial con-ditions prescribing the flow state att 0 and bound-ary conditions. In the time marching method thefinal steady state is achieved as the convergedsolution of unsteady calculations.

3 NUMERICAL METHOD

The governing equations are treated in conservativeform and discretized in time using the Euler implicitmethod leading to a set of non-linear finite differenceequations, which are solved using a Newtonprocedure. In stationary simulations, convergencehistory is accelerated by using local time steps

which are based on local stability criteria.Applying Greens theorem, the NavierStokes

equations can be rewritten in a differential form

@W

@t r F r G (2)

3.1 Inviscid fluxes

The convective (Euler) parts are discretized using athird-order accuracy, TVD-upwind finite volumescheme. When solving transport equations for turbu-lent quantities, a more stable first-order upwindscheme is used. The TVD scheme is based on theMUSCL-type of upwind scheme [7] which consists

of a projection stage and an evolution stage. In theprojection stage, left and right states at each controlvolume interface are determined by extrapolating thenode values of the conservative variables towardsthe control volume interface. In the evolution stagethe inviscid flux is evaluated by solving the Riemannproblem between left and right states using Roesapproximate Riemann solver [2].

3.2 Viscous fluxes

In order to construct the numerical viscous flux

vector at the control volume interfaces, it is neces-sary to evaluate first-order derivatives of the velocitycomponents, the speed of sound, and the turbulentquantities, which is done in a central-differencesmanner, using Greens theorem.

36 Y Mei and A Guha



3/13

3.3 Roe approximate Riemann solver

Consider Roes flux difference splitting of the inviscidflux vector

F(WL,WR) 12 F(WL)F(WR)

12

j~AA(WL,WR)

j (WL

WR)

(3)

wherej ~AA(WL,WR)j denotes the standard Roe matrix.It can be shown that this scheme is equivalent tothe first-order finite volume upwind cell vertexscheme based on a dual mesh. There are many differ-ent ways to achieve higher-order accuracy. In this

work, a scheme of higher-order accuracy is obtainedby using upwind-biased interpolations of the solu-tion WL,WR via the MUSCL approach. Therefore,the flux function is shown as

FLR 12 FLFR j ~AA(WL ,WR )j (WLWR ) (4)

where

FL F(WL ), FR F(WR )

and WL and WR are constructed by the MUSCL

scheme as

WL WLSL

4(1kSL) DL(1kSL)(WLWR)

WR WRSR

4(1kSR)DR(1kSR)(WRWL)

whereSis the flux limiter of von Albabda

Si2Di (ujui)1

(Di )2 (ujui)2 1

Sj2Dj (ujui)1

(Dj )2 (ujui)2 1

with 1!0.The forward and backward difference operators

are given by

DL 2WL, DR 2WR

The parameter k is chosen to control the degreeof approximation. In this paper, k 1/3, which cor-responds to a third-order upwind-biased scheme.

When k 0 it is a Fromm scheme, and k21 corres-ponds to a second-order upwind-biased scheme.

One of the strategies is to develop a multi-dimensional limiter for modifying high-order finitevolume schemes in such a way that the resultingmethod is monotonic, thus avoiding the creation ofunphysical oscillations in the numerical solution

and so improving the robustness of the algorithm.One commonly used approach is the slope limiting(MUSCL) technique of van Leer, in which the limiteris applied in a geometric manner to the gradients of apiecewise linear reconstruction of the solution tocreate a monotonic scheme.

3.4 Boundary condition

In the present scheme, phantom cells are used tohandle inlet and outlet boundaries. According tothe theory of characteristics, flow angle, total pres-sure, total temperature, and isentropic relations areused at the subsonic axial inlet, whereas all variablesare prescribed at the supersonic inlet. At the sub-sonic axial outlet the average value of the staticpressure is prescribed, whereas all variables arecalculated at the supersonic axial outlet. For Eulerequations, the slippery adiabatic or isothermal con-dition is applied on solid walls, and for NS equationsthe non-slip condition is used instead. All details areshown in Fig. 1.

1. Adiabatic impermeable solid wall: u.n 0 forinviscid flow computation; u 0 and @T/@n 0for viscous flow computation.

2. Periodic boundaries. Periodic boundaries are notnecessary if the whole bladerow is modelled,but, in practice, computer resources are limited,

and only one blade passage of the whole flowdomain is modelled to save computational time.This treatment is sufficient since flow data canbe transferred directly between periodic boun-daries, and the periodic boundary condition isprescribed on artificial cuts 2.

Fig. 1 Nomenclature for cascade of profiles SE1050

Implicit numerical simulation of transonic flow through turbine cascades 37


3.3 Roe approximate Riemann solver

Consider Roes flux difference splitting of the inviscidflux vector

F(WL,WR) 12 F(WL)F(WR)

12

j~AA(WL,WR)

j (WL

WR)

(3)

wherej ~AA(WL,WR)j denotes the standard Roe matrix.It can be shown that this scheme is equivalent tothe first-order finite volume upwind cell vertexscheme based on a dual mesh. There are many differ-ent ways to achieve higher-order accuracy. In this

work, a scheme of higher-order accuracy is obtainedby using upwind-biased interpolations of the solu-tion WL,WR via the MUSCL approach. Therefore,the flux function is shown as

FLR 12 FLFR j ~AA(WL ,WR )j (WLWR ) (4)

where

FL F(WL ), FR F(WR )

and WL and WR are constructed by the MUSCL

scheme as

WL WLSL

4(1kSL) DL(1kSL)(WLWR)

WR WRSR

4(1kSR)DR(1kSR)(WRWL)

whereSis the flux limiter of von Albabda

Si2Di (ujui)1

(Di )2 (ujui)2 1

Sj2Dj (ujui)1

(Dj )2 (ujui)2 1

with 1!0.The forward and backward difference operators

are given by

DL 2WL, DR 2WR

The parameter k is chosen to control the degreeof approximation. In this paper, k 1/3, which cor-responds to a third-order upwind-biased scheme.

When k 0 it is a Fromm scheme, and k21 corres-ponds to a second-order upwind-biased scheme.

One of the strategies is to develop a multi-dimensional limiter for modifying high-order finitevolume schemes in such a way that the resultingmethod is monotonic, thus avoiding the creation ofunphysical oscillations in the numerical solution

and so improving the robustness of the algorithm.One commonly used approach is the slope limiting(MUSCL) technique of van Leer, in which the limiteris applied in a geometric manner to the gradients of apiecewise linear reconstruction of the solution tocreate a monotonic scheme.

3.4 Boundary condition

In the present scheme, phantom cells are used tohandle inlet and outlet boundaries. According tothe theory of characteristics, flow angle, total pres-sure, total temperature, and isentropic relations areused at the subsonic axial inlet, whereas all variablesare prescribed at the supersonic inlet. At the sub-sonic axial outlet the average value of the staticpressure is prescribed, whereas all variables arecalculated at the supersonic axial outlet. For Eulerequations, the slippery adiabatic or isothermal con-dition is applied on solid walls, and for NS equationsthe non-slip condition is used instead. All details areshown in Fig. 1.

1. Adiabatic impermeable solid wall: u.n 0 forinviscid flow computation; u 0 and @T/@n 0for viscous flow computation.

2. Periodic boundaries. Periodic boundaries are notnecessary if the whole bladerow is modelled,but, in practice, computer resources are limited,

and only one blade passage of the whole flowdomain is modelled to save computational time.This treatment is sufficient since flow data canbe transferred directly between periodic boun-daries, and the periodic boundary condition isprescribed on artificial cuts 2.

Fig. 1 Nomenclature for cascade of profiles SE1050




4/13

3. Inlet boundary. Total pressure p0, total tempera-ture T0, and the inflow angle of attack a (orincidence angle i) are specified as the inputconditions.

4. Outlet boundary. Outlet static pressurep2is speci-fied. One option in the numerical simulation is toset the static pressure at the outlet boundary as an

invariant given by the input condition; the otheroption is to apply non-reflection treatment onthe outlet boundary as it has been successfullyapplied on the farfield condition of the externalflow computation. This method is supported bynumerical results of Yao et al. [8]. Both of thesemethods are used in the present paper, and thenumerical results based on non-reflecting con-ditions are shown to be closer to that of the realflow (i.e. experimental results).

Given the incoming Mach number, M1

, and theangle of attack, a, in the external computation, therest of the farfield properties are fixed. This makesit easy to apply the Roe scheme on the farfieldboundary calculation simply by treating the farfieldphantom cells as the left-hand-side properties

while treating the real values as the right-hand-sideproperties. This strategy makes the method that isused to compute the fluxes contributed by theboundaries the same as the method used in interiorflux computation. The internal consistency of treat-ments results in a large CFL number in the implicit

algorithm and will be discussed later in this paper.Unfortunately, unlike in the external computation,the inlet and outlet boundary conditions areexpressed implicitly in the internal flow compu-tation. The total pressure P0, the total temperatureT0, and the incident angleiare fixed at the entrance,and the outlet static pressure p2 is fixed at the exit.This does not provide enough information tospecify the flow properties which are necessary forapplying the Roe scheme at the inlet and outletboundary. To overcome this problem and to main-

tain the advantage of the consistency property atthe same time, a two-step prediction correctionprocedure is implemented here.

1. Prediction. There are three inlet boundary con-ditions, p0, T0, and a, and the conservative varia-bles will be specified explicitly if given anotherextra independent condition such as the inletmach number, M1, or the inlet static pressure,p1. For convenience of calculation, M1 is chosenas the extra parameter to specify the predictedvalues at the inlet phantom cells with the isentro-

pic assumption here; M1can easily be calculatedfrom the local inlet boundary nodes. At theoutlet boundary it is found that it would causeserious reflection from the outlet boundary ifp2

were set as a constant at the outlet boundary

nodes. To avoid violation of the non-reflectionboundary condition, p2 at the outlet boundary iskept flexible while p2 at the phantom cells isfixed. There may be a loss of the total pressureafter passing through the cascade profile. p0.2and T0.2 at the right-hand-side of the boundary(phantom cells) are taken the same as p0.2 and

T0.2 at the left-hand-side, which are calculatedfrom Wn at the outlet boundary. This, together

with specified p2 at the right-hand-side, enablesthe Roe scheme to calculate all properties at theoutlet boundary.

2. Correction. After upgrading the conservative varia-bles from Wn to Wn(1=2), a correction should beimposed on the inlet boundary nodes to makethem satisfy the constant total pressure conditionand constant total temperature at the inlet boun-dary. This can easily be realized by calculating the

Mach number M1 from the conservative valuesWn(1=2) at the inlet boundary and then updatingWn1 fromM1,p0,T0, and the inflow anglea.

Usually, the Steger Warming scheme is used for thefarfield boundary condition. In this paper, the far-field boundary condition is formulated such that itis internally consistent with the Roe scheme. Forthis, equation (3) is rewritten with the subscript Rreplaced by1 (the farfield condition)

F(WL,W1)

12

F(WL)

F(W1)

12j ~AA(WL,W1)j (WLW1) (3a)

The same equation applies for inflow (subscript 1)and outflow (subscript 2) farfeld conditions. Laterit will be shown that this internally consistenttreatment of boundary conditions improves theconvergence significantly.

3.5 Implicit time integration

Equation (1) can be rewritten in semi-discrete formfor each control volume

Vi@W

@tRi (5)

where Vi is the area for two dimensions and thevolume for three dimensions of the dual mesh cell,and Riis the residual. The Euler implicit discretiza-tion and linearization of equations leads to

V

Dt @F

@W nD

W

n

Rn

(6)

whereDtis the time increment, DWn is the differencein the conservation variable between time levels nand n1, and @F=@W represents symbolically the

38 Y Mei and A Guha


3. Inlet boundary. Total pressure p0, total tempera-ture T0, and the inflow angle of attack a (orincidence angle i) are specified as the inputconditions.

4. Outlet boundary. Outlet static pressurep2is speci-fied. One option in the numerical simulation is toset the static pressure at the outlet boundary as an

invariant given by the input condition; the otheroption is to apply non-reflection treatment onthe outlet boundary as it has been successfullyapplied on the farfield condition of the externalflow computation. This method is supported bynumerical results of Yao et al. [8]. Both of thesemethods are used in the present paper, and thenumerical results based on non-reflecting con-ditions are shown to be closer to that of the realflow (i.e. experimental results).

Given the incoming Mach number, M1

, and theangle of attack, a, in the external computation, therest of the farfield properties are fixed. This makesit easy to apply the Roe scheme on the farfieldboundary calculation simply by treating the farfieldphantom cells as the left-hand-side properties

while treating the real values as the right-hand-sideproperties. This strategy makes the method that isused to compute the fluxes contributed by theboundaries the same as the method used in interiorflux computation. The internal consistency of treat-ments results in a large CFL number in the implicit

algorithm and will be discussed later in this paper.Unfortunately, unlike in the external computation,the inlet and outlet boundary conditions areexpressed implicitly in the internal flow compu-tation. The total pressure P0, the total temperatureT0, and the incident angleiare fixed at the entrance,and the outlet static pressure p2 is fixed at the exit.This does not provide enough information tospecify the flow properties which are necessary forapplying the Roe scheme at the inlet and outletboundary. To overcome this problem and to main-

tain the advantage of the consistency property atthe same time, a two-step prediction correctionprocedure is implemented here.

1. Prediction. There are three inlet boundary con-ditions, p0, T0, and a, and the conservative varia-bles will be specified explicitly if given anotherextra independent condition such as the inletmach number, M1, or the inlet static pressure,p1. For convenience of calculation, M1 is chosenas the extra parameter to specify the predictedvalues at the inlet phantom cells with the isentro-

pic assumption here; M1can easily be calculatedfrom the local inlet boundary nodes. At theoutlet boundary it is found that it would causeserious reflection from the outlet boundary ifp2

were set as a constant at the outlet boundary

nodes. To avoid violation of the non-reflectionboundary condition, p2 at the outlet boundary iskept flexible while p2 at the phantom cells isfixed. There may be a loss of the total pressureafter passing through the cascade profile. p0.2and T0.2 at the right-hand-side of the boundary(phantom cells) are taken the same as p0.2 and

T0.2 at the left-hand-side, which are calculatedfrom Wn at the outlet boundary. This, together

with specified p2 at the right-hand-side, enablesthe Roe scheme to calculate all properties at theoutlet boundary.

2. Correction. After upgrading the conservative varia-bles from Wn to Wn(1=2), a correction should beimposed on the inlet boundary nodes to makethem satisfy the constant total pressure conditionand constant total temperature at the inlet boun-dary. This can easily be realized by calculating the

Mach number M1 from the conservative valuesWn(1=2) at the inlet boundary and then updatingWn1 fromM1,p0,T0, and the inflow anglea.

Usually, the Steger Warming scheme is used for thefarfield boundary condition. In this paper, the far-field boundary condition is formulated such that itis internally consistent with the Roe scheme. Forthis, equation (3) is rewritten with the subscript Rreplaced by1 (the farfield condition)

F(WL,W1)

12

F(WL)

F(W1)

12j ~AA(WL,W1)j (WLW1) (3a)

The same equation applies for inflow (subscript 1)and outflow (subscript 2) farfeld conditions. Laterit will be shown that this internally consistenttreatment of boundary conditions improves theconvergence significantly.

3.5 Implicit time integration

Equation (1) can be rewritten in semi-discrete formfor each control volume

Vi@W

@tRi (5)

where Vi is the area for two dimensions and thevolume for three dimensions of the dual mesh cell,and Riis the residual. The Euler implicit discretiza-tion and linearization of equations leads to

V

Dt @F

@W nD

W

n

Rn

(6)

whereDtis the time increment, DWn is the differencein the conservation variable between time levels nand n1, and @F=@W represents symbolically the

38 Y Mei and A Guha



5/13

Jacobian matrix. The Gauss Seidel algorithm isapplied in solving the linear equations at each stepof time integration.

The present work makes novel use of the Roescheme to calculate the implicit Jacobian matrix(in addition to its standard use for space discretiza-tion). The inviscid flux Jacobian matrix is thus

written as

@FLR

@WL 1

2A(WL) j ~AAj,

@FLR

@WR 1

2A(WR) j ~AAj (7)

For the farfield boundary condition the Jacobianmatrix is

@F

@Wi 1

2A(Wi) j ~AA(Wi,W1)j (8)

Therefore, all computational nodes in the flowfield,including the boundary points, are discretized bythe Roe scheme.

In the Navier Stokes calculation, the boundaryconditions on the body correspond to no slip andthe velocities at the solid boundary are known as 0and should not be treated as variables on the LHSof the equations. A post-processor is utilized toimplement these boundary conditions by modifyingthe matrix terms in equation (6) to reflect appropri-

ately the desired boundary conditions. For clearerdemonstration of this procedure, a slightly expandedrepresentation of one of the rows in equation (6) isgiven by

A11 A12 A13 A14A21 A22 A23 A24A31 A32 A33 A34A41 A42 A43 A44

2664

3775

Dr

DruDrvDe

2664

3775

R1R2R3R4

2664

3775 (9)

where R represents both the residual and off-diagonal terms on the right-hand side of equation(6) and Aijrepresents the individual components ofone of the diagonal blocks in [A]. The density canbe determined from the continuity equation duringthe solution process from the first row of equation(9). However, the contribution to the continuityresidual along the boundary involves integrationaround the dual mesh surrounding the node and asegment of the body surface. The contribution fromthe surface, assuming zero velocity at the wall, isidentically zero. The second and third rows are

modified so that the solution of equation (9) main-tains a zero velocity at the nodes on the solid boun-daries. Furthermore, the fourth row is altered topreserve a constant temperature in the case of anisothermal wall boundary condition. The constant

wall temperature assumption is used to relatethe change in energy at the wall to the change indensity

De Twallg(g1)Dr

The resulting matrices now reflect the implemen-tation of the appropriate boundary conditions atthe wall and are given by Anderson and Bonhaus [9].

1. Adiabatic wall

A11 A12 A13 A140 1 0 00 0 1 0

A41 A42 A43 A44

2664

3775

Dr

DruDrvDe

2664

3775

R100

R4

2664

3775 (10)

2. Isothermal wall

A11 A12 A13 A140 1 0 00 0 1 0

Twall=g(g 1) 0 0 1

2664

3775

Dr

DruDrvDe

2664

3775

R1000

2664

3775

(11)

3.6 Turbulence model

The algebraic turbulence model and the two-equation model are used here.

3.6.1 BaldwinLomax algebraic turbulence model

The BaldwinLomax (BL) turbulence model is themixing length model owing to the algebraic relation-

ship which uses Prandtl mixing length theory fordetermination of the eddy viscosity. This model iseasy to implement and can provide accurate resultsfor simple cases. It is a two-layer algebraic eddyviscosity model in which mtis given by

mt (mt)innery4 ycrossover

(mt)outery. ycrossover

whereyis the normal distance from the wall and thesmallest value ofyat which values from the inner and

outer formulation are equal.The Prandtlvan Driest formulation is used in the

inner region

v(i)T rl2jr uj



Jacobian matrix. The Gauss Seidel algorithm isapplied in solving the linear equations at each stepof time integration.

The present work makes novel use of the Roescheme to calculate the implicit Jacobian matrix(in addition to its standard use for space discretiza-tion). The inviscid flux Jacobian matrix is thus

written as

@FLR

@WL 1

2A(WL) j ~AAj,

@FLR

@WR 1

2A(WR) j ~AAj (7)

For the farfield boundary condition the Jacobianmatrix is

@F

@Wi 1

2A(Wi) j ~AA(Wi,W1)j (8)

Therefore, all computational nodes in the flowfield,including the boundary points, are discretized bythe Roe scheme.

In the Navier Stokes calculation, the boundaryconditions on the body correspond to no slip andthe velocities at the solid boundary are known as 0and should not be treated as variables on the LHSof the equations. A post-processor is utilized toimplement these boundary conditions by modifyingthe matrix terms in equation (6) to reflect appropri-

ately the desired boundary conditions. For clearerdemonstration of this procedure, a slightly expandedrepresentation of one of the rows in equation (6) isgiven by

A11 A12 A13 A14A21 A22 A23 A24A31 A32 A33 A34A41 A42 A43 A44

2664

3775

Dr

DruDrvDe

2664

3775

R1R2R3R4

2664

3775 (9)

where R represents both the residual and off-diagonal terms on the right-hand side of equation(6) and Aijrepresents the individual components ofone of the diagonal blocks in [A]. The density canbe determined from the continuity equation duringthe solution process from the first row of equation(9). However, the contribution to the continuityresidual along the boundary involves integrationaround the dual mesh surrounding the node and asegment of the body surface. The contribution fromthe surface, assuming zero velocity at the wall, isidentically zero. The second and third rows are

modified so that the solution of equation (9) main-tains a zero velocity at the nodes on the solid boun-daries. Furthermore, the fourth row is altered topreserve a constant temperature in the case of anisothermal wall boundary condition. The constant

wall temperature assumption is used to relatethe change in energy at the wall to the change indensity

De Twallg(g1)Dr

The resulting matrices now reflect the implemen-tation of the appropriate boundary conditions atthe wall and are given by Anderson and Bonhaus [9].

1. Adiabatic wall

A11 A12 A13 A140 1 0 00 0 1 0

A41 A42 A43 A44

2664

3775

Dr

DruDrvDe

2664

3775

R100

R4

2664

3775 (10)

2. Isothermal wall

A11 A12 A13 A140 1 0 00 0 1 0

Twall=g(g 1) 0 0 1

2664

3775

Dr

DruDrvDe

2664

3775

R1000

2664

3775

(11)

3.6 Turbulence model

The algebraic turbulence model and the two-equation model are used here.

3.6.1 BaldwinLomax algebraic turbulence model

The BaldwinLomax (BL) turbulence model is themixing length model owing to the algebraic relation-

ship which uses Prandtl mixing length theory fordetermination of the eddy viscosity. This model iseasy to implement and can provide accurate resultsfor simple cases. It is a two-layer algebraic eddyviscosity model in which mtis given by

mt (mt)innery4 ycrossover

(mt)outery. ycrossover

whereyis the normal distance from the wall and thesmallest value ofyat which values from the inner and

outer formulation are equal.The Prandtlvan Driest formulation is used in the

inner region

v(i)T rl2jr uj




6/13

where

lKy(1e(y=A))

yyv

ffiffiffiffiffit0

r

r

and ris the density, u is the velocity of the field, theshear stress tangential to the wall

t0m@u

@y

w

andA is the van-Driest constant (A 26).In the outer region, the eddy viscosity is given by

(mt)outer0:0168bFwakeFkleb

whereFwake is determined from

Fwakemin ymaxFmax0:25ymaxU2dif=Fmax

the quantity Udif is the difference between themaximum and minimum total velocity in the velocityprofile

Udif

( ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiu2 v2p )max( ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiu2 v2p )minymax is the distance from the wall at which F(y)becomes maximum, and Fmax is the maximumvalue of the function F(y), which is given as

F(y)y(1ey=A)jr uj

The intermittency factor is given by

F 11

5:5(0:3y=ymax)

Generally, the BL model is designed on the basis ofstructured grid computation because it needs thenormal distance from the wall. To adapt it forunstructured grids, a modification to the originalBL model has been made. A turbulence referencegrid is used to supply the necessary length scale foreach cell. It is constructed by dividing the compu-tational domain into strips, or narrow bands, whichemanate from the body walls or some wake centre-lines. The width of each strip should be of the order

of 12 cell sizes. The control volumes whose nodesreside in one particular strip are treated as onegroup of nodes. In each group, the distance of eachnode to the end-wall or the wake centre-line can bemeasured, which, in turn, will be used as the

reference length in the turbulence model. Once thereference length for each cell is established, compu-tation of the turbulence model can proceed on theunstructured mesh without difficulty. For example,the vorticity at each node can be obtained directlyfrom the velocity gradient computed by the flowsolver. The peak of a functional distribution can be

located by searching the extremum points alongeach strip or within each group. The turbulenceeddy viscosity at each node can then be determinedeasily by the two-layer algebraic mode.

Ideally, the divided strips should cover the entirecomputational domain without overlapping eachother. However, for the flowfield with multiple

walls (the pressure and suction surfaces of theblades) or wake lines, regions of overlapping strips

will occur. This is equivalent to saying that the tur-bulence in these regions is influenced by multiple

turbulence sources. In this case, the turbulence com-putation within each group still proceeds as before,but a weighting procedure is employed to determinethe influence from each wall or wake line. A moresophisticated approach may take the distance-

weighted average as the final turbulence eddyviscosity.

3.6.2 Two-equation turbulence model(kvmodel)

The main field of application of NavierStokesmethods in aerodynamics will be for complex turbu-lent flows that cannot be treated by an inviscid orviscousinviscid interaction scheme. Examples aremassively separated flows, flows involving multiplelength scales, flows with three-dimensional separa-tion, and complex unsteady flows. In these flowsthe application of algebraic turbulence models suchas the Baldwin Lomax model becomes very compli-cated owing to the difficulty in defining an algebraiclength scale. It is obvious that the improvement ofnumerical methods must be accompanied with thedevelopment of more general turbulence modelsand their implementation in NavierStokes codes.

The most popular non-algebraic turbulencemodels are two-equation eddy viscosity models.These models solve two transport equations, gener-ally one for the turbulent kinetic energy and anotherone related to turbulence length (or time) scale.

Among the two-equation models, the kv turbu-lence model of Wilcox is one of the most successful.It solves one equation for the turbulence kineticenergy, k, and a second equation for the specific

turbulent dissipation rate, v. The model does notemploy damping functions and has straightforwardDirichlet boundary conditions. This leads to signifi-cant advantages in numerical stability. The standard

Wilcoxkvmodel is given as follows.

40 Y Mei and A Guha


where

lKy(1e(y=A))

yyv

ffiffiffiffiffit0

r

r

and ris the density, u is the velocity of the field, theshear stress tangential to the wall

t0m@u

@y

w

andA is the van-Driest constant (A 26).In the outer region, the eddy viscosity is given by

(mt)outer0:0168bFwakeFkleb

whereFwake is determined from

Fwakemin ymaxFmax0:25ymaxU2dif=Fmax

the quantity Udif is the difference between themaximum and minimum total velocity in the velocityprofile

Udif

( ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiu2 v2p )max( ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiu2 v2p )minymax is the distance from the wall at which F(y)becomes maximum, and Fmax is the maximumvalue of the function F(y), which is given as

F(y)y(1ey=A)jr uj

The intermittency factor is given by

F 11

5:5(0:3y=ymax)

Generally, the BL model is designed on the basis ofstructured grid computation because it needs thenormal distance from the wall. To adapt it forunstructured grids, a modification to the originalBL model has been made. A turbulence referencegrid is used to supply the necessary length scale foreach cell. It is constructed by dividing the compu-tational domain into strips, or narrow bands, whichemanate from the body walls or some wake centre-lines. The width of each strip should be of the order

of 12 cell sizes. The control volumes whose nodesreside in one particular strip are treated as onegroup of nodes. In each group, the distance of eachnode to the end-wall or the wake centre-line can bemeasured, which, in turn, will be used as the

reference length in the turbulence model. Once thereference length for each cell is established, compu-tation of the turbulence model can proceed on theunstructured mesh without difficulty. For example,the vorticity at each node can be obtained directlyfrom the velocity gradient computed by the flowsolver. The peak of a functional distribution can be

located by searching the extremum points alongeach strip or within each group. The turbulenceeddy viscosity at each node can then be determinedeasily by the two-layer algebraic mode.

Ideally, the divided strips should cover the entirecomputational domain without overlapping eachother. However, for the flowfield with multiple

walls (the pressure and suction surfaces of theblades) or wake lines, regions of overlapping strips

will occur. This is equivalent to saying that the tur-bulence in these regions is influenced by multiple

turbulence sources. In this case, the turbulence com-putation within each group still proceeds as before,but a weighting procedure is employed to determinethe influence from each wall or wake line. A moresophisticated approach may take the distance-

weighted average as the final turbulence eddyviscosity.

3.6.2 Two-equation turbulence model(kvmodel)

The main field of application of NavierStokesmethods in aerodynamics will be for complex turbu-lent flows that cannot be treated by an inviscid orviscousinviscid interaction scheme. Examples aremassively separated flows, flows involving multiplelength scales, flows with three-dimensional separa-tion, and complex unsteady flows. In these flowsthe application of algebraic turbulence models suchas the Baldwin Lomax model becomes very compli-cated owing to the difficulty in defining an algebraiclength scale. It is obvious that the improvement ofnumerical methods must be accompanied with thedevelopment of more general turbulence modelsand their implementation in NavierStokes codes.

The most popular non-algebraic turbulencemodels are two-equation eddy viscosity models.These models solve two transport equations, gener-ally one for the turbulent kinetic energy and anotherone related to turbulence length (or time) scale.

Among the two-equation models, the kv turbu-lence model of Wilcox is one of the most successful.It solves one equation for the turbulence kineticenergy, k, and a second equation for the specific

turbulent dissipation rate, v. The model does notemploy damping functions and has straightforwardDirichlet boundary conditions. This leads to signifi-cant advantages in numerical stability. The standard

Wilcoxkvmodel is given as follows.

40 Y Mei and A Guha



7/13

For the turbulence kinetic energy k transportequation

@rk

@t@ruik

@xitij

@ui

@xjbrvk @

@xj(m sk1mt)

@k

@xj

For the turbulent dissipation rate v transportequation

@rv

@t @ruiv

@xi g1

nttij

@ui

@xj b1r2k

@@xj

(msv1mt)@v

@xj

where the constants will be

sk10:5, sv10:5, b10:0750b0:09, k0:41, g1

b1b

sv1k2ffiffiffiffiffi

bp

The model has to be supplemented by the definitionof the eddy viscosity

ntmtrk

v

The turbulent stress tensor tij ru0iu0j is thengiven by

tijmt@ui@xj

@uj@xi

23

@uk@xk

dij

2

3rkdij

For the turbulent equations, the boundary condi-tions are needed. For the wall conditions, no slipconditions are used. The turbulent kinetic energykis zero, and vsatisfies the following equation [6]

v10 6nb1(Dy)

2

where Dyis the distance to the first point away fromthe wall. For freestream conditions, the turbulentkinetic energy k is set at 1027, and the turbulentcoefficient is 0.01 (non-dimensional), so for thefreestreamvRe=(M1m1)r1k1.

This model has been widely used. It is wall dis-tance free, which is very important for the unstruc-tured grid, and does not need damping functions in

the viscous sublayer. However, this model is sensitiveto the value specified for the freestream specificdissipation, v1. The turbulent coefficient changesgreatly, but for wall-bounded flow this is not verysignificant.

3.7 Computational grids

All computational unstructured grids are generatedby either the DelaunayVoronoi method (DVM) orthe advancing front method (AFM). Thin grids arenecessary to capture the flow details in the boundarylayer, and for this purpose the advancing layer

method is applied in generating skew meshesinside the boundary layer when the solver is consid-ering the effect of viscosity. Figures 2 and 3 are theunstructured grids generated for Euler equationcalculations. Figure 2 is the original grid generated

Fig. 2 Initial mesh around the SE1050 blade for the

Euler solver

Fig. 3 Computational grid for the Euler solver after

fivefold refinement



For the turbulence kinetic energy k transportequation

@rk

@t@ruik

@xitij

@ui

@xjbrvk @

@xj(m sk1mt)

@k

@xj

For the turbulent dissipation rate v transportequation

@rv

@t @ruiv

@xi g1

nttij

@ui

@xj b1r2k

@@xj

(msv1mt)@v

@xj

where the constants will be

sk10:5, sv10:5, b10:0750b0:09, k0:41, g1

b1b

sv1k2ffiffiffiffiffi

bp

The model has to be supplemented by the definitionof the eddy viscosity

ntmtrk

v

The turbulent stress tensor tij ru0iu0j is thengiven by

tijmt@ui@xj

@uj@xi

23

@uk@xk

dij

2

3rkdij

For the turbulent equations, the boundary condi-tions are needed. For the wall conditions, no slipconditions are used. The turbulent kinetic energykis zero, and vsatisfies the following equation [6]

v10 6nb1(Dy)

2

where Dyis the distance to the first point away fromthe wall. For freestream conditions, the turbulentkinetic energy k is set at 1027, and the turbulentcoefficient is 0.01 (non-dimensional), so for thefreestreamvRe=(M1m1)r1k1.

This model has been widely used. It is wall dis-tance free, which is very important for the unstruc-tured grid, and does not need damping functions in

the viscous sublayer. However, this model is sensitiveto the value specified for the freestream specificdissipation, v1. The turbulent coefficient changesgreatly, but for wall-bounded flow this is not verysignificant.

3.7 Computational grids

All computational unstructured grids are generatedby either the DelaunayVoronoi method (DVM) orthe advancing front method (AFM). Thin grids arenecessary to capture the flow details in the boundarylayer, and for this purpose the advancing layer

method is applied in generating skew meshesinside the boundary layer when the solver is consid-ering the effect of viscosity. Figures 2 and 3 are theunstructured grids generated for Euler equationcalculations. Figure 2 is the original grid generated

Fig. 2 Initial mesh around the SE1050 blade for the

Euler solver

Fig. 3 Computational grid for the Euler solver after

fivefold refinement




8/13

by DVM, and Fig. 3 is the grid after fivefoldrefinement.

4 NUMERICAL RESULTS

The final objective of the present research is to inves-

tigate the aerodynamic and thermal properties ofthe effect of homogeneous condensation flow onthe performance of the cascade. The focus here isupon single-phase flow through a two-dimensionalsteam turbine cascade. The transonic rotor turbineprofile cascade SE1050 [10], which was designed forthe last stage of a SKODA steam turbine, is chosenas the test case here. Extensive measurementsusing this profile have been made for a largenumber of operating conditions. A rigorous defi-nition of measured quantities is given in reference

[10]. This provides a useful comparison with numeri-cal computations. In addition, this cascade satisfiesthe velocity range used in modern gas turbines.

The physical experiment was performed underthree conditions including the design condition andtwo off-design conditions. The numerical experi-ments were performed at the design condition forthe following data: angle of attacka19:38(corres-ponding to the incident angle i08), total pressurep096 748 Pa, total temperature T0296:65K,Reynolds number Re 1.48106, and isentropicoutlet Mach number M2,is

1:198.

4.1 Inviscid computation

These computations were carried out to validate theinviscid part of the solver. The original mesh used forthe present computation consists of 2605 nodes, of

which 222 nodes lie on the blade surface, and 4508cells, as shown in Fig. 2. It takes 22 time steps to con-verge to 25 orders with the GaussSeidel implicitalgorithm. The fine grid after fivefold self-adaptationcontains 8427 nodes and 16 062 cells, see Fig. 3. Theadvantage of the interior consistency of the inviscidsolver is obvious: the CFL number can be as largeas 10 000 and still the scheme maintains stability.Since the computational error is proportional toO(Dt), too large a CFL number would result in alarge numerical error. Therefore, the maximum CFLnumber is set at 200 during the calculation to achievea compromise between efficiency and accuracy.

The calculated surface pressure distribution alongthe chord is plotted in Fig. 4. The pressure coefficientCpis defined asp/p0. From Fig. 4 it can be seen that

the pressure changes slowly along the pressure sidewhile it changes rapidly along the suction side.The pressure downstream of the reflection shock

wave that is generated from the trailing edge of thesuction side of the blade corresponds to the outlet

static pressure p2 and remains approximatelyconstant until the arc of the trailing edge is reached.The convergence histories of the computationsare compared in Fig. 5. The first test case is a Roescheme carried out on a coarse grid, the second is

a Roe scheme on a fine grid, and the third is an Osherscheme on the same fine grid. The Roe method isapplied at the inflow and outflow boundaries in all

Fig. 4 Surface pressure distribution along the chord of

SE1050, calculated by the present Euler solver

Fig. 5 Convergence history of the computations on

coarse and fine grids for various schemes for

internal nodes and boundary nodes

42 Y Mei and A Guha


by DVM, and Fig. 3 is the grid after fivefoldrefinement.

4 NUMERICAL RESULTS

The final objective of the present research is to inves-

tigate the aerodynamic and thermal properties ofthe effect of homogeneous condensation flow onthe performance of the cascade. The focus here isupon single-phase flow through a two-dimensionalsteam turbine cascade. The transonic rotor turbineprofile cascade SE1050 [10], which was designed forthe last stage of a SKODA steam turbine, is chosenas the test case here. Extensive measurementsusing this profile have been made for a largenumber of operating conditions. A rigorous defi-nition of measured quantities is given in reference

[10]. This provides a useful comparison with numeri-cal computations. In addition, this cascade satisfiesthe velocity range used in modern gas turbines.

The physical experiment was performed underthree conditions including the design condition andtwo off-design conditions. The numerical experi-ments were performed at the design condition forthe following data: angle of attacka19:38(corres-ponding to the incident angle i08), total pressurep096 748 Pa, total temperature T0296:65K,Reynolds number Re 1.48106, and isentropicoutlet Mach number M2,is

1:198.

4.1 Inviscid computation

These computations were carried out to validate theinviscid part of the solver. The original mesh used forthe present computation consists of 2605 nodes, of

which 222 nodes lie on the blade surface, and 4508cells, as shown in Fig. 2. It takes 22 time steps to con-verge to 25 orders with the GaussSeidel implicitalgorithm. The fine grid after fivefold self-adaptationcontains 8427 nodes and 16 062 cells, see Fig. 3. Theadvantage of the interior consistency of the inviscidsolver is obvious: the CFL number can be as largeas 10 000 and still the scheme maintains stability.Since the computational error is proportional toO(Dt), too large a CFL number would result in alarge numerical error. Therefore, the maximum CFLnumber is set at 200 during the calculation to achievea compromise between efficiency and accuracy.

The calculated surface pressure distribution alongthe chord is plotted in Fig. 4. The pressure coefficientCpis defined asp/p0. From Fig. 4 it can be seen that

the pressure changes slowly along the pressure sidewhile it changes rapidly along the suction side.The pressure downstream of the reflection shock

wave that is generated from the trailing edge of thesuction side of the blade corresponds to the outlet

static pressure p2 and remains approximatelyconstant until the arc of the trailing edge is reached.The convergence histories of the computationsare compared in Fig. 5. The first test case is a Roescheme carried out on a coarse grid, the second is

a Roe scheme on a fine grid, and the third is an Osherscheme on the same fine grid. The Roe method isapplied at the inflow and outflow boundaries in all

Fig. 4 Surface pressure distribution along the chord of

SE1050, calculated by the present Euler solver

Fig. 5 Convergence history of the computations on

coarse and fine grids for various schemes for

internal nodes and boundary nodes

42 Y Mei and A Guha



9/13

three test cases. The RoeRoe method on a coarsegrid shows the best convergence history because ofthe large artificial viscosity brought by the largegrid. Comparison between the Roe Roe andOsher Roe methods on the self-adaptive gridconfirms that internal consistency of the solver willprovide a more efficient computation.

Safarik suggested that the relative velocity l beused as a standard parameter in the internal flowcomputation. Here, l is defined as the ratio ofvelocity and the critical speed of sound

l uffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi2g=(g1)RT0

p

A comparison of the computed surface relativevelocity distribution with the experimental data ofStastny and Safarik [10] is shown in Fig. 6. In thisfigure, the results denoted by circles have beenobtained with the inviscid computation while thosedenoted by the squares are obtained from the experi-ment. Since the real velocities on the solid wall arezero, the relative velocity data in the experiment

were obtained at the edge of the boundary layer.The agreement between the computations and the

experiment is good over most of the blade, althoughan overexpansion near the trailing edge is evidentin the computations which does not appear in theexperiment. The round trailing edge of the SE1050blade profile causes this difficulty in the computation

of Euler equations. Denton and Dawes [11] foundthat, with the coarse grid, it is not necessaryto apply the Kutta condition explicitly and the

numerical viscosity in the solution will automaticallydamp the flow and make it leave the trailing edgesmoothly. However, with the refinement of the gridsafter several self-adaptive processes, more nodes areset on the trailing edge automatically for accuratelycapture of the dramatic changes. The flow then triesto approach a stagnation point on the trailing edge

circle and the point where the Kutta conditionshould be imposed is not well defined. Then, thepredicted flow is found to accelerate to high speedat the start of the trailing edge circle on both pressureand suction surfaces. From the numerical point ofview, the high-speed flows from the pressure sideand the suction side clash somewhere at the trailingedge with opposite directions and stop at the stag-nation point all of a sudden. The dramatic change inthe physical properties occurs within a few gridsoccupying a short distance along the trailing edge

and destabilizes the numerical robustness.To obtain a desirable solution, Denton rec-

ommended that the trailing edge of the blade beextended along the streamline, so the round shapeof the trailing edge is changed into a sharp onemaking the flow leave the trailing edge smoothly.This technology is known as CUSP, and it simplifiesthe complexity of the computation and does notinfluence the flow significantly. The CUSP shouldbe unloaded, so that it only introduces blockage

without exerting tangential force on the flow. Unfor-

tunately, there is no theoretical instruction abouthow to create such a CUSP, and experience isneeded in its proper implementation.

Figures 7 and 8 represent the Mach contours ofthe numerical and experimental results respectively.Figure 8 shows the interferometric picture with linesof constant Mach number. Several shock waves,boundary layers, wakes, and expansion waves areobserved. (The figure presents the frame aroundtrailing edges of a cascade of profiles.) The fishtailshock waves are seen from both the pressure andsuction sides of the trailing edge circle. The shock

wave generated from the pressure side is verystrong and hits the suction side of its neighbouringblade. It can be seen from a comparison of Figs 7and 8 that the solver captured exact positions ofthe reflection shock wave.

4.2 Viscous computation

Many (20 in the present calculation) layers ofnodes are distributed in the boundary layer tosimulate the real flow more accurately. Both the

Baldwin Lomax algebraic turbulence model andthe two-equation kv model are incorporated inthe developed solver. The initial computational gridis plotted in Fig. 9, while the surface pressure distri-bution and the Mach number contour are shown in

Fig. 6 Comparison between the experimental and

numerical relative surface velocityl along the

chord of SE1050 (M2,is 1.198,a 19.38)



three test cases. The RoeRoe method on a coarsegrid shows the best convergence history because ofthe large artificial viscosity brought by the largegrid. Comparison between the Roe Roe andOsher Roe methods on the self-adaptive gridconfirms that internal consistency of the solver willprovide a more efficient computation.

Safarik suggested that the relative velocity l beused as a standard parameter in the internal flowcomputation. Here, l is defined as the ratio ofvelocity and the critical speed of sound

l uffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi2g=(g1)RT0

p

A comparison of the computed surface relativevelocity distribution with the experimental data ofStastny and Safarik [10] is shown in Fig. 6. In thisfigure, the results denoted by circles have beenobtained with the inviscid computation while thosedenoted by the squares are obtained from the experi-ment. Since the real velocities on the solid wall arezero, the relative velocity data in the experiment

were obtained at the edge of the boundary layer.The agreement between the computations and the

experiment is good over most of the blade, althoughan overexpansion near the trailing edge is evidentin the computations which does not appear in theexperiment. The round trailing edge of the SE1050blade profile causes this difficulty in the computation

of Euler equations. Denton and Dawes [11] foundthat, with the coarse grid, it is not necessaryto apply the Kutta condition explicitly and the

numerical viscosity in the solution will automaticallydamp the flow and make it leave the trailing edgesmoothly. However, with the refinement of the gridsafter several self-adaptive processes, more nodes areset on the trailing edge automatically for accuratelycapture of the dramatic changes. The flow then triesto approach a stagnation point on the trailing edge

circle and the point where the Kutta conditionshould be imposed is not well defined. Then, thepredicted flow is found to accelerate to high speedat the start of the trailing edge circle on both pressureand suction surfaces. From the numerical point ofview, the high-speed flows from the pressure sideand the suction side clash somewhere at the trailingedge with opposite directions and stop at the stag-nation point all of a sudden. The dramatic change inthe physical properties occurs within a few gridsoccupying a short distance along the trailing edge

and destabilizes the numerical robustness.To obtain a desirable solution, Denton rec-

ommended that the trailing edge of the blade beextended along the streamline, so the round shapeof the trailing edge is changed into a sharp onemaking the flow leave the trailing edge smoothly.This technology is known as CUSP, and it simplifiesthe complexity of the computation and does notinfluence the flow significantly. The CUSP shouldbe unloaded, so that it only introduces blockage

without exerting tangential force on the flow. Unfor-

tunately, there is no theoretical instruction abouthow to create such a CUSP, and experience isneeded in its proper implementation.

Figures 7 and 8 represent the Mach contours ofthe numerical and experimental results respectively.Figure 8 shows the interferometric picture with linesof constant Mach number. Several shock waves,boundary layers, wakes, and expansion waves areobserved. (The figure presents the frame aroundtrailing edges of a cascade of profiles.) The fishtailshock waves are seen from both the pressure andsuction sides of the trailing edge circle. The shock

wave generated from the pressure side is verystrong and hits the suction side of its neighbouringblade. It can be seen from a comparison of Figs 7and 8 that the solver captured exact positions ofthe reflection shock wave.

4.2 Viscous computation

Many (20 in the present calculation) layers ofnodes are distributed in the boundary layer tosimulate the real flow more accurately. Both the

Baldwin Lomax algebraic turbulence model andthe two-equation kv model are incorporated inthe developed solver. The initial computational gridis plotted in Fig. 9, while the surface pressure distri-bution and the Mach number contour are shown in

Fig. 6 Comparison between the experimental and

numerical relative surface velocityl along the

chord of SE1050 (M2,is 1.198,a 19.38)




10/13

Figs 10 and 11 respectively. The operating conditionis the same as the inviscid computation, with theReynolds number as an additional parameter: inci-dent angle i 19.38, M2,is1:198, Re 1.48106,total pressure P0 99 805 Pa, and total temperatureT0 300 K.

Comparing the pressure distribution of the inviscidresult in Fig. 4 with the viscous result in Fig. 10, it isinteresting to see that the oscillation at the trailingedge disappeared. In the viscous computation, thevelocity on the solid wall is fixed at zero, so it reducesnon-physical oscillation and enforces stability ofthe calculation. Comparison between Figs 4 and 10shows that the shock wave moves towards the leadingedge, which results from boundary layer blockage.

Different self-adaptive sensors are applied in thegrid refinement procedure for efficiently detectingthe positions of the shock wave and the wake. Theshock is characterized as an abrupt jump in pressure,density, and other flow variables. Therefore, the den-sity gradient is chosen as the shock wave sensor toindicate the position of the shock wave. Unfortu-

nately, this sensor does not detect the wake, becauseneither the pressure nor the density changes signifi-cantly in the wake. To capture the wake, which isdominated by the viscosity, various rotation-relatedsensors are constructed and tested as follows.

1. Primitive rotation variable v. The primitiverotation variable, v, is chosen as the first wakesensor. As expected, the wake is captured in the

Fig. 8 Interferometric picture of flow in the SE1050

cascade (a 0, M2,is 1.198) (courtesy of

P. Safarik)

Fig. 7 Mach contour on the SE1050 blade cascade,

calculated by the present Euler solver (a 0,

M2,is 1.198)

Fig. 9 Initial computational grid around the SE1050

blade for the NS solver

44 Y Mei and A Guha


Figs 10 and 11 respectively. The operating conditionis the same as the inviscid computation, with theReynolds number as an additional parameter: inci-dent angle i 19.38, M2,is1:198, Re 1.48106,total pressure P0 99 805 Pa, and total temperatureT0 300 K.

Comparing the pressure distribution of the inviscidresult in Fig. 4 with the viscous result in Fig. 10, it isinteresting to see that the oscillation at the trailingedge disappeared. In the viscous computation, thevelocity on the solid wall is fixed at zero, so it reducesnon-physical oscillation and enforces stability ofthe calculation. Comparison between Figs 4 and 10shows that the shock wave moves towards the leadingedge, which results from boundary layer blockage.

Different self-adaptive sensors are applied in thegrid refinement procedure for efficiently detectingthe positions of the shock wave and the wake. Theshock is characterized as an abrupt jump in pressure,density, and other flow variables. Therefore, the den-sity gradient is chosen as the shock wave sensor toindicate the position of the shock wave. Unfortu-

nately, this sensor does not detect the wake, becauseneither the pressure nor the density changes signifi-cantly in the wake. To capture the wake, which isdominated by the viscosity, various rotation-relatedsensors are constructed and tested as follows.

1. Primitive rotation variable v. The primitiverotation variable, v, is chosen as the first wakesensor. As expected, the wake is captured in the

Fig. 8 Interferometric picture of flow in the SE1050

cascade (a 0, M2,is 1.198) (courtesy of

P. Safarik)

Fig. 7 Mach contour on the SE1050 blade cascade,

calculated by the present Euler solver (a 0,

M2,is 1.198)

Fig. 9 Initial computational grid around the SE1050

blade for the NS solver

44 Y Mei and A Guha



11/13

first three generations of self-adaptation. How-ever, unfortunately, the new nodes are addedsuch that the dense region of grids extends like a

fork within the wake, the effect becoming moreprominent as more generations of self-adaptationare employed. The reason for this non-uniformdistribution is that the transverse gradient ofspeed near the centre-line of the wake decreasesin the axial downstream direction. This meansthat the rotation variable v near the centre-lineof the wake decreases more rapidly than that ofthe other area in the wake.

2. New wake sensorv=(u 2 V2). To get rid of thefork and predict an accurate position of the

wake, a new wake sensor is created. Consideringthat the velocities on the centre-line are relativelysmall compared with those on the other area ofthe wake, the new wake-sensitive sensor isdevised as v=(u2 v2). The numerical tests inthe present study showed that this sensor ismore effective for self-adaptation of grids withinthe wake.

Figure 11 shows the computational mesh after threerounds of refinement, two of them based on the

shock sensor and one based on the wake sensor.The pressure distribution along the blade surface isplotted in Fig. 12. A comparison with Fig. 10 showsthat the shock wave is much sharper after the gridrefinement. Figure 13 plots the Mach contour

based on the refined grid and provides an instantview of the wake of the flow.

The numerical results of the inviscid and viscousflow computations are listed in Table 1 and com-pared with experimental results [12]. Five numericalsolutions from the present study are included: Euler(on a coarse grid), Euler (on a fine grid), NavierStokes (NS) (laminar), NS (BL turbulent model),and NS (kvturbulent model).

Fig. 10 Pressure distribution along the chord of the

SE1050 blade with akvmodel based on the

initial grid

Fig. 11 Refined grid for N S computation

Fig. 12 Pressure distribution along the blade chord

based on the refined grid, calculated by the

present NS solver with akvmodel



first three generations of self-adaptation. How-ever, unfortunately, the new nodes are addedsuch that the dense region of grids extends like a

fork within the wake, the effect becoming moreprominent as more generations of self-adaptationare employed. The reason for this non-uniformdistribution is that the transverse gradient ofspeed near the centre-line of the wake decreasesin the axial downstream direction. This meansthat the rotation variable v near the centre-lineof the wake decreases more rapidly than that ofthe other area in the wake.

2. New wake sensorv=(u 2 V2). To get rid of thefork and predict an accurate position of the

wake, a new wake sensor is created. Consideringthat the velocities on the centre-line are relativelysmall compared with those on the other area ofthe wake, the new wake-sensitive sensor isdevised as v=(u2 v2). The numerical tests inthe present study showed that this sensor ismore effective for self-adaptation of grids withinthe wake.

Figure 11 shows the computational mesh after threerounds of refinement, two of them based on the

shock sensor and one based on the wake sensor.The pressure distribution along the blade surface isplotted in Fig. 12. A comparison with Fig. 10 showsthat the shock wave is much sharper after the gridrefinement. Figure 13 plots the Mach contour

based on the refined grid and provides an instantview of the wake of the flow.

The numerical results of the inviscid and viscousflow computations are listed in Table 1 and com-pared with experimental results [12]. Five numericalsolutions from the present study are included: Euler(on a coarse grid), Euler (on a fine grid), NavierStokes (NS) (laminar), NS (BL turbulent model),and NS (kvturbulent model).

Fig. 10 Pressure distribution along the chord of the

SE1050 blade with akvmodel based on the

initial grid

Fig. 11 Refined grid for N S computation

Fig. 12 Pressure distribution along the blade chord

based on the refined grid, calculated by the

present NS solver with akvmodel




12/13

The energy loss coefficient,j, is defined [12] as

j100 1 l22

l22,is

!

wherel2,isis the isentropic outlet relative speed andl2is the real outlet relative speed. The jvalues calcu-lated in different computations are listed in Table 1.

As expected the viscous results give a higher energyloss than do the inviscid results. The loss coefficientpredicted by the inviscid computation depends onthe grid size: j 2.1 per cent in the fine grid andj 2.6 per cent in the coarse grid. When the gridsize is small, the shocks are sharper and the numeri-cal viscosity is lower. For the viscous computations,the kv model predicts the highest energy loss

(which is also the closest to the experiment result),

and jof the laminar flow is the smallest because ofthe absence of turbulence viscosity.

5 CONCLUSION

Computational solvers of Euler and Navier Stokes

equations have been developed. A hybrid scheme isused for spatial discretization, where the inviscidfluxes are discretized using a finite volume method

while the viscous fluxes are calculated by centraldifferences. A MUSCL-type approach is used forachieving higher-order accuracy. The effects of theturbulent stress terms in the Reynolds-averagedNavierStokes equations have been studied with twodifferent models: an algebraic turbulence model(BaldwinLomax model) and a two-equation turbu-lence model (kv model). The calculations have

been carried out on unstructured grids. The initialgrids for inviscid and viscous computations are gener-ated by the advancing front method (AFM) andDelaunayVeronoi method (DVM), while the self-adaptive procedure is based on the DelaunayVeronoimethod (DVM). A new rotation-based sensor has beendevised to capture the wake region more accurately.

The numerical solvers have been applied to thetransonic rotor turbine profile cascade SE1050. Thenumerical results of the inviscid and viscous flowcomputations are listed in Table 1 and compared

with experimental results [12]. Five numerical solu-tions from the present study are included: Euler (ona coarse grid), Euler (on a fine grid), NS (laminar), NS(BL turbulent model), and NS (kvturbulent model).Comparisons with experiment include relative velo-city distributions and Mach number contours. Ingeneral, the comparisons with experiment are good.

In the present work, a new method is developedsuch that the calculation of the flux, the boundaryconditions, and the implicit Jacobian all conform tothe Roe scheme. This provides internal consistencyand improves the convergence significantly. Manynumerical tests have been performed with differentexpressions for the inlet and outlet boundaries. Thetraditional Steger Warming scheme and the newRoe scheme are tested, combined with differentinviscid flux solvers: the Osher scheme and the Roescheme. The present numerical study (e.g. Fig. 5)shows that the new treatment of the boundary con-ditions at the inlet and outlet boundaries improvesthe computational efficiency significantly.

Work is at hand to augment the Euler and Navier Stokes solvers to simulate non-equilibrium wet

steam flow through turbine cascades. References[13] and [14] present an Eulerian Lagrangianapproach for wet steam calculations with manynovel features (e.g. accurate coupling of unsteadyfluid dynamics, a new averaging procedure for

Fig. 13 Mach contour on the SE1050 blade cascade

with akvmodel based on the refined grid

Table 1 Comparison of the experimental and various

numerical results

InletMachnumber,M1

Exitangle,b2(deg)

Energylosscoefficient,j(%)

Experimental [12] 0.375 29.80 4.6Computational [12] 0.329 32.02 Not availableEuler (on a coarse grid) 0.329 31.94 2.5

Euler (on a fine grid) 0.329 31.91 2.1NS (laminar) 0.328 31.80 2.6NS (BL turbulent

model)0.327 31.95 2.9

NS (kvturbulentmodel)

0.328 31.93 3.1

46 Y Mei and A Guha


The energy loss coefficient,j, is defined [12] as

j100 1 l22

l22,is

!

wherel2,isis the isentropic outlet relative speed andl2is the real outlet relative speed. The jvalues calcu-lated in different computations are listed in Table 1.

As expected the viscous results give a higher energyloss than do the inviscid results. The loss coefficientpredicted by the inviscid computation depends onthe grid size: j 2.1 per cent in the fine grid andj 2.6 per cent in the coarse grid. When the gridsize is small, the shocks are sharper and the numeri-cal viscosity is lower. For the viscous computations,the kv model predicts the highest energy loss

(which is also the closest to the experiment result),

and jof the laminar flow is the smallest because ofthe absence of turbulence viscosity.

5 CONCLUSION

Computational solvers of Euler and Navier Stokes

equations have been developed. A hybrid scheme isused for spatial discretization, where the inviscidfluxes are discretized using a finite volume method

while the viscous fluxes are calculated by centraldifferences. A MUSCL-type approach is used forachieving higher-order accuracy. The effects of theturbulent stress terms in the Reynolds-averagedNavierStokes equations have been studied with twodifferent models: an algebraic turbulence model(BaldwinLomax model) and a two-equation turbu-lence model (kv model). The calculations have

been carried out on unstructured grids. The initialgrids for inviscid and viscous computations are gener-ated by the advancing front method (AFM) andDelaunayVeronoi method (DVM), while the self-adaptive procedure is based on the DelaunayVeronoimethod (DVM). A new rotation-based sensor has beendevised to capture the wake region more accurately.

The numerical solvers have been applied to thetransonic rotor turbine profile cascade SE1050. Thenumerical results of the inviscid and viscous flowcomputations are listed in Table 1 and compared

with experimental results [12]. Five numerical solu-tions from the present study are included: Euler (ona coarse grid), Euler (on a fine grid), NS (laminar), NS(BL turbulent model), and NS (kvturbulent model).Comparisons with experiment include relative velo-city distributions and Mach number contours. Ingeneral, the comparisons with experiment are good.

In the present work, a new method is developedsuch that the calculation of the flux, the boundaryconditions, and the implicit Jacobian all conform tothe Roe scheme. This provides internal consistencyand improves the convergence significantly. Manynumerical tests have been performed with differentexpressions for the inlet and outlet boundaries. Thetraditional Steger Warming scheme and the newRoe scheme are tested, combined with differentinviscid flux solvers: the Osher scheme and the Roescheme. The present numerical study (e.g. Fig. 5)shows that the new treatment of the boundary con-ditions at the inlet and outlet boundaries improvesthe computational efficiency significantly.

Work is at hand to augment the Euler and Navier Stokes solvers to simulate non-equilibrium wet

steam flow through turbine cascades. References[13] and [14] present an Eulerian Lagrangianapproach for wet steam calculations with manynovel features (e.g. accurate coupling of unsteadyfluid dynamics, a new averaging procedure for

Fig. 13 Mach contour on the SE1050 blade cascade

with akvmodel based on the refined grid

Table 1 Comparison of the experimental and various

numerical results

InletMachnumber,M1

Exitangle,b2(deg)

Energylosscoefficient,j(%)

Experimental [12] 0.375 29.80 4.6Computational [12] 0.329 32.02 Not availableEuler (on a coarse grid) 0.329 31.94 2.5

Euler (on a fine grid) 0.329 31.91 2.1NS (laminar) 0.328 31.80 2.6NS (BL turbulent

model)0.327 31.95 2.9

NS (kvturbulentmodel)

0.328 31.93 3.1

46 Y Mei and A Guha



13/13

retaining a poly-dispersed droplet spectrum etc.),whereas the two-phase solvers under developmentwill use an EulerianEulerian scheme.

ACKNOWLEDGEMENTS

The authors wish to express their gratitude toP. Safarik who provided the experimental data andhelpful comments. YM was supported by a Univer-sity of Bristol Scholarship and an ORS award.

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AIAA paper 78257, 1978.6 Wilcox, D. C. Turbulence Modelling for CFD, 1993 (DCW

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ASME paper 90-GT-313, 1990.11 Denton, J. D. and Dawes, W. N. Computational fluid

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implicit turbine cascade cfd solver

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